85675
What is the minimum volume of the parallelopiped?
1 \(\frac{3 \sqrt{3}-2}{3 \sqrt{3}}\) cubic unit
2 \(\frac{3 \sqrt{3}-1}{3 \sqrt{3}}\) cubic unit
3 \(\frac{3 \sqrt{3}-2}{\sqrt{3}}\) cubic unit
4 None of the above
Explanation:
(A) : Volume of paralleopiped- \(\left|\begin{array}{lll}1 & m & 1 \\ 0 & 1 & m \\ m & 0 & 1\end{array}\right|=1-m\left(0-m^2\right)+(0-m)\) \(=1-m\left(0-m^{2}\right)+(0-m) \tag{i}\) \(=m^{3}-m+1\) Let, \(f(m)=m^{3}-m+1\) \(\mathrm{f}^{\prime}(\mathrm{m})=3 \mathrm{~m}^{2}-1=0\) \(\Rightarrow \mathrm{m}= \pm \frac{1}{\sqrt{3}}\) Minimum occurs at \(\mathrm{m}=\frac{1}{\sqrt{3}}\) Put \(\mathrm{m}=\frac{1}{\sqrt{3}}\) in equation (i), we get- \(=\frac{1}{3 \sqrt{3}}-\frac{1}{\sqrt{3}}+1=\frac{3 \sqrt{3}-2}{3 \sqrt{3}} \text { cubic unit }\)
SCRA-2010
Application of Derivatives
85676
What is the left derivative of the function \(\mathbf{f}(\mathbf{x})=\max \left(|\mathbf{x}|, \sqrt{|\mathbf{x}|}, \mathbf{x}^{3}\right)\) at \(\mathrm{x}=-1\) ?
1 1
2 3
3 \(-1 / 2\)
4 -1
Explanation:
(D) : Given, \(f(x)=\max \left(|x|, \sqrt{|x|}, x^{3}\right)\) It is clear from the graph that \(x\lt -1\) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}|, \mathrm{x}\lt 0\) \(=-\mathrm{x}\) L.H.D. \(=\mathrm{f}^{\prime}(\mathrm{x})=-1\) \(\Rightarrow \mathrm{f}^{\prime}(-1)=-1\)
SCRA-2010
Application of Derivatives
85677
For the function \(f(x)=\int_{0}^{x} \frac{\operatorname{sint}}{t} d t\), which one of the following is correct?
1 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is even
2 Minimum occurs at \(x=n \pi\), where \(n\) is odd
3 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
4 None of the above
Explanation:
(C) : Given, \(f(x)= \int_0^x \frac{\sin t}{t} d t\) \(f^{\prime}(x)=1 \cdot \frac{\sin x}{x}-0 \cdot\left(\frac{\sin 0}{0}\right)=\frac{\sin x}{x}=0\) \(\sin x=0\) \(x=n \pi\) \(f^{\prime \prime}(x)=\frac{x \cos x-\sin x}{x^2}\left\{\begin{array}{l} f^{\prime \prime}(x)\lt 0 \text { for max ima } \\ f^{\prime \prime}(x)>0 \text { for minima } \end{array}\right.\) \(n=\text { odd } \Rightarrow x=\pi\) \(f^{\prime \prime}(x)=\frac{\cos \pi-\sin \pi}{\pi^2} \Rightarrow-\frac{1}{\pi^2}\lt 0\) \(n=e v e n \Rightarrow x^{\prime}=2 \pi\) \(f^{\prime \prime}(x)=\frac{2 \pi}{\pi^2}=\frac{2}{\pi}>0\) So, Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
SCRA-2010
Application of Derivatives
85678
The maximum value of \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) is
1 10
2 -10
3 14
4 None of these
Explanation:
(A) : Given, \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) \(=5 \cos \theta+3\left[\cos \theta \cdot \cos 60^{\circ}-\sin \theta \sin 60^{\circ}\right]+3\) \(=5 \cos \theta+3\left[\frac{\cos \theta}{2}-\frac{\sqrt{3}}{2} \sin \theta\right]+3\) \(=5 \cos \theta+\frac{3 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) \(=\frac{13 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) This is a type of \(\mathrm{a} \cos \theta-\mathrm{b} \sin \theta+3\) Where, \(\mathrm{a}=\frac{13}{2}, \mathrm{~b}=\frac{3 \sqrt{3}}{2}\) Then, maximum value of this type of expression \(=\sqrt{a^{2}+b^{2}}+3\) So, after putting value of \(a\) and \(b\), we get - \(\sqrt{\frac{169}{4}+\frac{27}{4}}+3=\sqrt{\frac{196}{4}}+3=\sqrt{49}+3\) \(=7+3=10=\max \text { value }\)
CG PET-2010
Application of Derivatives
85679
Let \(f(x)=a-(x-3)^{8 / 9}\), then maxima of \(f(x)\) is
1 3
2 \(a-3\)
3 a
4 None of these
Explanation:
(C) : Given \(\mathrm{f}(\mathrm{x})=\mathrm{a}-(\mathrm{x}-3)^{8 /}\) Then, \(y=a-(x-3)^{8 / 9}\) Differentiating both side, we gat - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0-\frac{8}{9}(\mathrm{x}-3)^{\frac{8}{9}-1}\) \(\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{8}{9}(\mathrm{x}-3)^{\frac{-1}{9}}\) For maxima of \(f(x)\) is - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0 \Rightarrow-\frac{8}{9} \frac{1}{(\mathrm{x}-3)^{1 / 9}}=0\) At, \(x=3, \frac{d y}{d x}\) is not define So, the maximum of \(f(x)\) is - \(\left.\mathrm{f}(\mathrm{x})\right|_{\mathrm{x}=3}=\mathrm{f}(3)=\mathrm{a}\)
85675
What is the minimum volume of the parallelopiped?
1 \(\frac{3 \sqrt{3}-2}{3 \sqrt{3}}\) cubic unit
2 \(\frac{3 \sqrt{3}-1}{3 \sqrt{3}}\) cubic unit
3 \(\frac{3 \sqrt{3}-2}{\sqrt{3}}\) cubic unit
4 None of the above
Explanation:
(A) : Volume of paralleopiped- \(\left|\begin{array}{lll}1 & m & 1 \\ 0 & 1 & m \\ m & 0 & 1\end{array}\right|=1-m\left(0-m^2\right)+(0-m)\) \(=1-m\left(0-m^{2}\right)+(0-m) \tag{i}\) \(=m^{3}-m+1\) Let, \(f(m)=m^{3}-m+1\) \(\mathrm{f}^{\prime}(\mathrm{m})=3 \mathrm{~m}^{2}-1=0\) \(\Rightarrow \mathrm{m}= \pm \frac{1}{\sqrt{3}}\) Minimum occurs at \(\mathrm{m}=\frac{1}{\sqrt{3}}\) Put \(\mathrm{m}=\frac{1}{\sqrt{3}}\) in equation (i), we get- \(=\frac{1}{3 \sqrt{3}}-\frac{1}{\sqrt{3}}+1=\frac{3 \sqrt{3}-2}{3 \sqrt{3}} \text { cubic unit }\)
SCRA-2010
Application of Derivatives
85676
What is the left derivative of the function \(\mathbf{f}(\mathbf{x})=\max \left(|\mathbf{x}|, \sqrt{|\mathbf{x}|}, \mathbf{x}^{3}\right)\) at \(\mathrm{x}=-1\) ?
1 1
2 3
3 \(-1 / 2\)
4 -1
Explanation:
(D) : Given, \(f(x)=\max \left(|x|, \sqrt{|x|}, x^{3}\right)\) It is clear from the graph that \(x\lt -1\) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}|, \mathrm{x}\lt 0\) \(=-\mathrm{x}\) L.H.D. \(=\mathrm{f}^{\prime}(\mathrm{x})=-1\) \(\Rightarrow \mathrm{f}^{\prime}(-1)=-1\)
SCRA-2010
Application of Derivatives
85677
For the function \(f(x)=\int_{0}^{x} \frac{\operatorname{sint}}{t} d t\), which one of the following is correct?
1 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is even
2 Minimum occurs at \(x=n \pi\), where \(n\) is odd
3 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
4 None of the above
Explanation:
(C) : Given, \(f(x)= \int_0^x \frac{\sin t}{t} d t\) \(f^{\prime}(x)=1 \cdot \frac{\sin x}{x}-0 \cdot\left(\frac{\sin 0}{0}\right)=\frac{\sin x}{x}=0\) \(\sin x=0\) \(x=n \pi\) \(f^{\prime \prime}(x)=\frac{x \cos x-\sin x}{x^2}\left\{\begin{array}{l} f^{\prime \prime}(x)\lt 0 \text { for max ima } \\ f^{\prime \prime}(x)>0 \text { for minima } \end{array}\right.\) \(n=\text { odd } \Rightarrow x=\pi\) \(f^{\prime \prime}(x)=\frac{\cos \pi-\sin \pi}{\pi^2} \Rightarrow-\frac{1}{\pi^2}\lt 0\) \(n=e v e n \Rightarrow x^{\prime}=2 \pi\) \(f^{\prime \prime}(x)=\frac{2 \pi}{\pi^2}=\frac{2}{\pi}>0\) So, Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
SCRA-2010
Application of Derivatives
85678
The maximum value of \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) is
1 10
2 -10
3 14
4 None of these
Explanation:
(A) : Given, \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) \(=5 \cos \theta+3\left[\cos \theta \cdot \cos 60^{\circ}-\sin \theta \sin 60^{\circ}\right]+3\) \(=5 \cos \theta+3\left[\frac{\cos \theta}{2}-\frac{\sqrt{3}}{2} \sin \theta\right]+3\) \(=5 \cos \theta+\frac{3 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) \(=\frac{13 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) This is a type of \(\mathrm{a} \cos \theta-\mathrm{b} \sin \theta+3\) Where, \(\mathrm{a}=\frac{13}{2}, \mathrm{~b}=\frac{3 \sqrt{3}}{2}\) Then, maximum value of this type of expression \(=\sqrt{a^{2}+b^{2}}+3\) So, after putting value of \(a\) and \(b\), we get - \(\sqrt{\frac{169}{4}+\frac{27}{4}}+3=\sqrt{\frac{196}{4}}+3=\sqrt{49}+3\) \(=7+3=10=\max \text { value }\)
CG PET-2010
Application of Derivatives
85679
Let \(f(x)=a-(x-3)^{8 / 9}\), then maxima of \(f(x)\) is
1 3
2 \(a-3\)
3 a
4 None of these
Explanation:
(C) : Given \(\mathrm{f}(\mathrm{x})=\mathrm{a}-(\mathrm{x}-3)^{8 /}\) Then, \(y=a-(x-3)^{8 / 9}\) Differentiating both side, we gat - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0-\frac{8}{9}(\mathrm{x}-3)^{\frac{8}{9}-1}\) \(\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{8}{9}(\mathrm{x}-3)^{\frac{-1}{9}}\) For maxima of \(f(x)\) is - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0 \Rightarrow-\frac{8}{9} \frac{1}{(\mathrm{x}-3)^{1 / 9}}=0\) At, \(x=3, \frac{d y}{d x}\) is not define So, the maximum of \(f(x)\) is - \(\left.\mathrm{f}(\mathrm{x})\right|_{\mathrm{x}=3}=\mathrm{f}(3)=\mathrm{a}\)
85675
What is the minimum volume of the parallelopiped?
1 \(\frac{3 \sqrt{3}-2}{3 \sqrt{3}}\) cubic unit
2 \(\frac{3 \sqrt{3}-1}{3 \sqrt{3}}\) cubic unit
3 \(\frac{3 \sqrt{3}-2}{\sqrt{3}}\) cubic unit
4 None of the above
Explanation:
(A) : Volume of paralleopiped- \(\left|\begin{array}{lll}1 & m & 1 \\ 0 & 1 & m \\ m & 0 & 1\end{array}\right|=1-m\left(0-m^2\right)+(0-m)\) \(=1-m\left(0-m^{2}\right)+(0-m) \tag{i}\) \(=m^{3}-m+1\) Let, \(f(m)=m^{3}-m+1\) \(\mathrm{f}^{\prime}(\mathrm{m})=3 \mathrm{~m}^{2}-1=0\) \(\Rightarrow \mathrm{m}= \pm \frac{1}{\sqrt{3}}\) Minimum occurs at \(\mathrm{m}=\frac{1}{\sqrt{3}}\) Put \(\mathrm{m}=\frac{1}{\sqrt{3}}\) in equation (i), we get- \(=\frac{1}{3 \sqrt{3}}-\frac{1}{\sqrt{3}}+1=\frac{3 \sqrt{3}-2}{3 \sqrt{3}} \text { cubic unit }\)
SCRA-2010
Application of Derivatives
85676
What is the left derivative of the function \(\mathbf{f}(\mathbf{x})=\max \left(|\mathbf{x}|, \sqrt{|\mathbf{x}|}, \mathbf{x}^{3}\right)\) at \(\mathrm{x}=-1\) ?
1 1
2 3
3 \(-1 / 2\)
4 -1
Explanation:
(D) : Given, \(f(x)=\max \left(|x|, \sqrt{|x|}, x^{3}\right)\) It is clear from the graph that \(x\lt -1\) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}|, \mathrm{x}\lt 0\) \(=-\mathrm{x}\) L.H.D. \(=\mathrm{f}^{\prime}(\mathrm{x})=-1\) \(\Rightarrow \mathrm{f}^{\prime}(-1)=-1\)
SCRA-2010
Application of Derivatives
85677
For the function \(f(x)=\int_{0}^{x} \frac{\operatorname{sint}}{t} d t\), which one of the following is correct?
1 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is even
2 Minimum occurs at \(x=n \pi\), where \(n\) is odd
3 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
4 None of the above
Explanation:
(C) : Given, \(f(x)= \int_0^x \frac{\sin t}{t} d t\) \(f^{\prime}(x)=1 \cdot \frac{\sin x}{x}-0 \cdot\left(\frac{\sin 0}{0}\right)=\frac{\sin x}{x}=0\) \(\sin x=0\) \(x=n \pi\) \(f^{\prime \prime}(x)=\frac{x \cos x-\sin x}{x^2}\left\{\begin{array}{l} f^{\prime \prime}(x)\lt 0 \text { for max ima } \\ f^{\prime \prime}(x)>0 \text { for minima } \end{array}\right.\) \(n=\text { odd } \Rightarrow x=\pi\) \(f^{\prime \prime}(x)=\frac{\cos \pi-\sin \pi}{\pi^2} \Rightarrow-\frac{1}{\pi^2}\lt 0\) \(n=e v e n \Rightarrow x^{\prime}=2 \pi\) \(f^{\prime \prime}(x)=\frac{2 \pi}{\pi^2}=\frac{2}{\pi}>0\) So, Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
SCRA-2010
Application of Derivatives
85678
The maximum value of \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) is
1 10
2 -10
3 14
4 None of these
Explanation:
(A) : Given, \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) \(=5 \cos \theta+3\left[\cos \theta \cdot \cos 60^{\circ}-\sin \theta \sin 60^{\circ}\right]+3\) \(=5 \cos \theta+3\left[\frac{\cos \theta}{2}-\frac{\sqrt{3}}{2} \sin \theta\right]+3\) \(=5 \cos \theta+\frac{3 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) \(=\frac{13 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) This is a type of \(\mathrm{a} \cos \theta-\mathrm{b} \sin \theta+3\) Where, \(\mathrm{a}=\frac{13}{2}, \mathrm{~b}=\frac{3 \sqrt{3}}{2}\) Then, maximum value of this type of expression \(=\sqrt{a^{2}+b^{2}}+3\) So, after putting value of \(a\) and \(b\), we get - \(\sqrt{\frac{169}{4}+\frac{27}{4}}+3=\sqrt{\frac{196}{4}}+3=\sqrt{49}+3\) \(=7+3=10=\max \text { value }\)
CG PET-2010
Application of Derivatives
85679
Let \(f(x)=a-(x-3)^{8 / 9}\), then maxima of \(f(x)\) is
1 3
2 \(a-3\)
3 a
4 None of these
Explanation:
(C) : Given \(\mathrm{f}(\mathrm{x})=\mathrm{a}-(\mathrm{x}-3)^{8 /}\) Then, \(y=a-(x-3)^{8 / 9}\) Differentiating both side, we gat - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0-\frac{8}{9}(\mathrm{x}-3)^{\frac{8}{9}-1}\) \(\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{8}{9}(\mathrm{x}-3)^{\frac{-1}{9}}\) For maxima of \(f(x)\) is - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0 \Rightarrow-\frac{8}{9} \frac{1}{(\mathrm{x}-3)^{1 / 9}}=0\) At, \(x=3, \frac{d y}{d x}\) is not define So, the maximum of \(f(x)\) is - \(\left.\mathrm{f}(\mathrm{x})\right|_{\mathrm{x}=3}=\mathrm{f}(3)=\mathrm{a}\)
NEET Test Series from KOTA - 10 Papers In MS WORD
WhatsApp Here
Application of Derivatives
85675
What is the minimum volume of the parallelopiped?
1 \(\frac{3 \sqrt{3}-2}{3 \sqrt{3}}\) cubic unit
2 \(\frac{3 \sqrt{3}-1}{3 \sqrt{3}}\) cubic unit
3 \(\frac{3 \sqrt{3}-2}{\sqrt{3}}\) cubic unit
4 None of the above
Explanation:
(A) : Volume of paralleopiped- \(\left|\begin{array}{lll}1 & m & 1 \\ 0 & 1 & m \\ m & 0 & 1\end{array}\right|=1-m\left(0-m^2\right)+(0-m)\) \(=1-m\left(0-m^{2}\right)+(0-m) \tag{i}\) \(=m^{3}-m+1\) Let, \(f(m)=m^{3}-m+1\) \(\mathrm{f}^{\prime}(\mathrm{m})=3 \mathrm{~m}^{2}-1=0\) \(\Rightarrow \mathrm{m}= \pm \frac{1}{\sqrt{3}}\) Minimum occurs at \(\mathrm{m}=\frac{1}{\sqrt{3}}\) Put \(\mathrm{m}=\frac{1}{\sqrt{3}}\) in equation (i), we get- \(=\frac{1}{3 \sqrt{3}}-\frac{1}{\sqrt{3}}+1=\frac{3 \sqrt{3}-2}{3 \sqrt{3}} \text { cubic unit }\)
SCRA-2010
Application of Derivatives
85676
What is the left derivative of the function \(\mathbf{f}(\mathbf{x})=\max \left(|\mathbf{x}|, \sqrt{|\mathbf{x}|}, \mathbf{x}^{3}\right)\) at \(\mathrm{x}=-1\) ?
1 1
2 3
3 \(-1 / 2\)
4 -1
Explanation:
(D) : Given, \(f(x)=\max \left(|x|, \sqrt{|x|}, x^{3}\right)\) It is clear from the graph that \(x\lt -1\) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}|, \mathrm{x}\lt 0\) \(=-\mathrm{x}\) L.H.D. \(=\mathrm{f}^{\prime}(\mathrm{x})=-1\) \(\Rightarrow \mathrm{f}^{\prime}(-1)=-1\)
SCRA-2010
Application of Derivatives
85677
For the function \(f(x)=\int_{0}^{x} \frac{\operatorname{sint}}{t} d t\), which one of the following is correct?
1 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is even
2 Minimum occurs at \(x=n \pi\), where \(n\) is odd
3 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
4 None of the above
Explanation:
(C) : Given, \(f(x)= \int_0^x \frac{\sin t}{t} d t\) \(f^{\prime}(x)=1 \cdot \frac{\sin x}{x}-0 \cdot\left(\frac{\sin 0}{0}\right)=\frac{\sin x}{x}=0\) \(\sin x=0\) \(x=n \pi\) \(f^{\prime \prime}(x)=\frac{x \cos x-\sin x}{x^2}\left\{\begin{array}{l} f^{\prime \prime}(x)\lt 0 \text { for max ima } \\ f^{\prime \prime}(x)>0 \text { for minima } \end{array}\right.\) \(n=\text { odd } \Rightarrow x=\pi\) \(f^{\prime \prime}(x)=\frac{\cos \pi-\sin \pi}{\pi^2} \Rightarrow-\frac{1}{\pi^2}\lt 0\) \(n=e v e n \Rightarrow x^{\prime}=2 \pi\) \(f^{\prime \prime}(x)=\frac{2 \pi}{\pi^2}=\frac{2}{\pi}>0\) So, Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
SCRA-2010
Application of Derivatives
85678
The maximum value of \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) is
1 10
2 -10
3 14
4 None of these
Explanation:
(A) : Given, \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) \(=5 \cos \theta+3\left[\cos \theta \cdot \cos 60^{\circ}-\sin \theta \sin 60^{\circ}\right]+3\) \(=5 \cos \theta+3\left[\frac{\cos \theta}{2}-\frac{\sqrt{3}}{2} \sin \theta\right]+3\) \(=5 \cos \theta+\frac{3 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) \(=\frac{13 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) This is a type of \(\mathrm{a} \cos \theta-\mathrm{b} \sin \theta+3\) Where, \(\mathrm{a}=\frac{13}{2}, \mathrm{~b}=\frac{3 \sqrt{3}}{2}\) Then, maximum value of this type of expression \(=\sqrt{a^{2}+b^{2}}+3\) So, after putting value of \(a\) and \(b\), we get - \(\sqrt{\frac{169}{4}+\frac{27}{4}}+3=\sqrt{\frac{196}{4}}+3=\sqrt{49}+3\) \(=7+3=10=\max \text { value }\)
CG PET-2010
Application of Derivatives
85679
Let \(f(x)=a-(x-3)^{8 / 9}\), then maxima of \(f(x)\) is
1 3
2 \(a-3\)
3 a
4 None of these
Explanation:
(C) : Given \(\mathrm{f}(\mathrm{x})=\mathrm{a}-(\mathrm{x}-3)^{8 /}\) Then, \(y=a-(x-3)^{8 / 9}\) Differentiating both side, we gat - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0-\frac{8}{9}(\mathrm{x}-3)^{\frac{8}{9}-1}\) \(\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{8}{9}(\mathrm{x}-3)^{\frac{-1}{9}}\) For maxima of \(f(x)\) is - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0 \Rightarrow-\frac{8}{9} \frac{1}{(\mathrm{x}-3)^{1 / 9}}=0\) At, \(x=3, \frac{d y}{d x}\) is not define So, the maximum of \(f(x)\) is - \(\left.\mathrm{f}(\mathrm{x})\right|_{\mathrm{x}=3}=\mathrm{f}(3)=\mathrm{a}\)
85675
What is the minimum volume of the parallelopiped?
1 \(\frac{3 \sqrt{3}-2}{3 \sqrt{3}}\) cubic unit
2 \(\frac{3 \sqrt{3}-1}{3 \sqrt{3}}\) cubic unit
3 \(\frac{3 \sqrt{3}-2}{\sqrt{3}}\) cubic unit
4 None of the above
Explanation:
(A) : Volume of paralleopiped- \(\left|\begin{array}{lll}1 & m & 1 \\ 0 & 1 & m \\ m & 0 & 1\end{array}\right|=1-m\left(0-m^2\right)+(0-m)\) \(=1-m\left(0-m^{2}\right)+(0-m) \tag{i}\) \(=m^{3}-m+1\) Let, \(f(m)=m^{3}-m+1\) \(\mathrm{f}^{\prime}(\mathrm{m})=3 \mathrm{~m}^{2}-1=0\) \(\Rightarrow \mathrm{m}= \pm \frac{1}{\sqrt{3}}\) Minimum occurs at \(\mathrm{m}=\frac{1}{\sqrt{3}}\) Put \(\mathrm{m}=\frac{1}{\sqrt{3}}\) in equation (i), we get- \(=\frac{1}{3 \sqrt{3}}-\frac{1}{\sqrt{3}}+1=\frac{3 \sqrt{3}-2}{3 \sqrt{3}} \text { cubic unit }\)
SCRA-2010
Application of Derivatives
85676
What is the left derivative of the function \(\mathbf{f}(\mathbf{x})=\max \left(|\mathbf{x}|, \sqrt{|\mathbf{x}|}, \mathbf{x}^{3}\right)\) at \(\mathrm{x}=-1\) ?
1 1
2 3
3 \(-1 / 2\)
4 -1
Explanation:
(D) : Given, \(f(x)=\max \left(|x|, \sqrt{|x|}, x^{3}\right)\) It is clear from the graph that \(x\lt -1\) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}|, \mathrm{x}\lt 0\) \(=-\mathrm{x}\) L.H.D. \(=\mathrm{f}^{\prime}(\mathrm{x})=-1\) \(\Rightarrow \mathrm{f}^{\prime}(-1)=-1\)
SCRA-2010
Application of Derivatives
85677
For the function \(f(x)=\int_{0}^{x} \frac{\operatorname{sint}}{t} d t\), which one of the following is correct?
1 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is even
2 Minimum occurs at \(x=n \pi\), where \(n\) is odd
3 Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
4 None of the above
Explanation:
(C) : Given, \(f(x)= \int_0^x \frac{\sin t}{t} d t\) \(f^{\prime}(x)=1 \cdot \frac{\sin x}{x}-0 \cdot\left(\frac{\sin 0}{0}\right)=\frac{\sin x}{x}=0\) \(\sin x=0\) \(x=n \pi\) \(f^{\prime \prime}(x)=\frac{x \cos x-\sin x}{x^2}\left\{\begin{array}{l} f^{\prime \prime}(x)\lt 0 \text { for max ima } \\ f^{\prime \prime}(x)>0 \text { for minima } \end{array}\right.\) \(n=\text { odd } \Rightarrow x=\pi\) \(f^{\prime \prime}(x)=\frac{\cos \pi-\sin \pi}{\pi^2} \Rightarrow-\frac{1}{\pi^2}\lt 0\) \(n=e v e n \Rightarrow x^{\prime}=2 \pi\) \(f^{\prime \prime}(x)=\frac{2 \pi}{\pi^2}=\frac{2}{\pi}>0\) So, Maximum occurs at \(\mathrm{x}=\mathrm{n} \pi\), where \(\mathrm{n}\) is odd
SCRA-2010
Application of Derivatives
85678
The maximum value of \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) is
1 10
2 -10
3 14
4 None of these
Explanation:
(A) : Given, \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) \(=5 \cos \theta+3\left[\cos \theta \cdot \cos 60^{\circ}-\sin \theta \sin 60^{\circ}\right]+3\) \(=5 \cos \theta+3\left[\frac{\cos \theta}{2}-\frac{\sqrt{3}}{2} \sin \theta\right]+3\) \(=5 \cos \theta+\frac{3 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) \(=\frac{13 \cos \theta}{2}-\frac{3 \sqrt{3}}{2} \sin \theta+3\) This is a type of \(\mathrm{a} \cos \theta-\mathrm{b} \sin \theta+3\) Where, \(\mathrm{a}=\frac{13}{2}, \mathrm{~b}=\frac{3 \sqrt{3}}{2}\) Then, maximum value of this type of expression \(=\sqrt{a^{2}+b^{2}}+3\) So, after putting value of \(a\) and \(b\), we get - \(\sqrt{\frac{169}{4}+\frac{27}{4}}+3=\sqrt{\frac{196}{4}}+3=\sqrt{49}+3\) \(=7+3=10=\max \text { value }\)
CG PET-2010
Application of Derivatives
85679
Let \(f(x)=a-(x-3)^{8 / 9}\), then maxima of \(f(x)\) is
1 3
2 \(a-3\)
3 a
4 None of these
Explanation:
(C) : Given \(\mathrm{f}(\mathrm{x})=\mathrm{a}-(\mathrm{x}-3)^{8 /}\) Then, \(y=a-(x-3)^{8 / 9}\) Differentiating both side, we gat - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0-\frac{8}{9}(\mathrm{x}-3)^{\frac{8}{9}-1}\) \(\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{8}{9}(\mathrm{x}-3)^{\frac{-1}{9}}\) For maxima of \(f(x)\) is - \(\frac{\mathrm{dy}}{\mathrm{dx}}=0 \Rightarrow-\frac{8}{9} \frac{1}{(\mathrm{x}-3)^{1 / 9}}=0\) At, \(x=3, \frac{d y}{d x}\) is not define So, the maximum of \(f(x)\) is - \(\left.\mathrm{f}(\mathrm{x})\right|_{\mathrm{x}=3}=\mathrm{f}(3)=\mathrm{a}\)
