85671
If \(a x^{2}+b x+4\) attains its minimum value -1 at \(\mathbf{x}=1\), then the values of \(\mathbf{a}\) and \(b\) are respectively
1 \(5,-10\)
2 \(5,-5\)
3 5,5
4 \(10,-5\)
Explanation:
(A): \(f(x)=a x^{2}+b x+4\) \(f^{\prime}(x)=2 a x+b\) For local minima, \(\mathrm{f}^{\prime}(\mathrm{x})=0\) \(2 a x+b=0\) \(x=-\frac{b}{2 a}\) At \(x=1 f^{\prime}(1)=2 a+b=0\) \(f(1)=a(1)^{2}+b(1)+4 \tag{i}\) \(-1=a+b+4\) \(a+b+5=0 \tag{ii}\) Equation (i) and (ii), we get \(a=5, \quad b=-10\)
BCECE-2016
Application of Derivatives
85672
The maximum value of \(12 \sin \theta-9 \sin ^{2} \theta\) is :
1 3
2 4
3 5
4 none of these
Explanation:
(B) : \(12 \sin \theta-9 \sin ^{2} \theta\) \(=4-4+12 \sin \theta-9 \sin ^{2} \theta=4-(2-3 \sin \theta)^{2}\) For maximum value, \((2-3 \sin \theta)^{2}\) gives the minimum value, \((2-3 \sin \theta)^{2} =0\) \(\sin \theta =\frac{2}{3}\) \(\therefore\) Maximum value, at \(\sin \theta=\frac{2}{3}\) \(12 \sin \theta-9 \sin ^{2} \theta\) \(=12 \times \frac{2}{3}-9 \times\left(\frac{2}{3}\right)^{2}=8-4=4\)
BCECE-2004
Application of Derivatives
85673
The function \(\frac{a \sin x+b \cos x}{c \sin x+d \cos x}, x \in R\) attains neither maximum nor minimum if,
85674
Which one of the following is correct in respect of the function \(f(x)=5 \log x-24 x+32 x^{2} ; x>0\) ?
1 It has neither a point of maximum nor a point of minimum
2 It has a point of maximum, but no point of minimum
3 It has a point of minimum, but no point of maximum.
4 It has a point of maximum and a point of minimum
Explanation:
(A) : Given, \(f(x)=5 \log x-24 x+32 x^{2} ; x>0\) Diff. w.r.t. \(\mathrm{x}\) we get- \(f^{\prime}(x)=\frac{5}{x}-24+64 x\) Put, \(\mathrm{f}^{\prime}(\mathrm{x})=0\) \(\frac{5}{x}-24+64 x=0\) \(\Rightarrow 64 x^{2}-24 x+5=0\) Now, \(D=(-24)^{2}-4 \times 5 \times 64\) \(D=576-1280\) \(D=-704\lt 0\) Now real value of \(x\) exist. Hence, it has neither a point of maxima nor a point of minima.
NEET Test Series from KOTA - 10 Papers In MS WORD
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Application of Derivatives
85671
If \(a x^{2}+b x+4\) attains its minimum value -1 at \(\mathbf{x}=1\), then the values of \(\mathbf{a}\) and \(b\) are respectively
1 \(5,-10\)
2 \(5,-5\)
3 5,5
4 \(10,-5\)
Explanation:
(A): \(f(x)=a x^{2}+b x+4\) \(f^{\prime}(x)=2 a x+b\) For local minima, \(\mathrm{f}^{\prime}(\mathrm{x})=0\) \(2 a x+b=0\) \(x=-\frac{b}{2 a}\) At \(x=1 f^{\prime}(1)=2 a+b=0\) \(f(1)=a(1)^{2}+b(1)+4 \tag{i}\) \(-1=a+b+4\) \(a+b+5=0 \tag{ii}\) Equation (i) and (ii), we get \(a=5, \quad b=-10\)
BCECE-2016
Application of Derivatives
85672
The maximum value of \(12 \sin \theta-9 \sin ^{2} \theta\) is :
1 3
2 4
3 5
4 none of these
Explanation:
(B) : \(12 \sin \theta-9 \sin ^{2} \theta\) \(=4-4+12 \sin \theta-9 \sin ^{2} \theta=4-(2-3 \sin \theta)^{2}\) For maximum value, \((2-3 \sin \theta)^{2}\) gives the minimum value, \((2-3 \sin \theta)^{2} =0\) \(\sin \theta =\frac{2}{3}\) \(\therefore\) Maximum value, at \(\sin \theta=\frac{2}{3}\) \(12 \sin \theta-9 \sin ^{2} \theta\) \(=12 \times \frac{2}{3}-9 \times\left(\frac{2}{3}\right)^{2}=8-4=4\)
BCECE-2004
Application of Derivatives
85673
The function \(\frac{a \sin x+b \cos x}{c \sin x+d \cos x}, x \in R\) attains neither maximum nor minimum if,
85674
Which one of the following is correct in respect of the function \(f(x)=5 \log x-24 x+32 x^{2} ; x>0\) ?
1 It has neither a point of maximum nor a point of minimum
2 It has a point of maximum, but no point of minimum
3 It has a point of minimum, but no point of maximum.
4 It has a point of maximum and a point of minimum
Explanation:
(A) : Given, \(f(x)=5 \log x-24 x+32 x^{2} ; x>0\) Diff. w.r.t. \(\mathrm{x}\) we get- \(f^{\prime}(x)=\frac{5}{x}-24+64 x\) Put, \(\mathrm{f}^{\prime}(\mathrm{x})=0\) \(\frac{5}{x}-24+64 x=0\) \(\Rightarrow 64 x^{2}-24 x+5=0\) Now, \(D=(-24)^{2}-4 \times 5 \times 64\) \(D=576-1280\) \(D=-704\lt 0\) Now real value of \(x\) exist. Hence, it has neither a point of maxima nor a point of minima.
85671
If \(a x^{2}+b x+4\) attains its minimum value -1 at \(\mathbf{x}=1\), then the values of \(\mathbf{a}\) and \(b\) are respectively
1 \(5,-10\)
2 \(5,-5\)
3 5,5
4 \(10,-5\)
Explanation:
(A): \(f(x)=a x^{2}+b x+4\) \(f^{\prime}(x)=2 a x+b\) For local minima, \(\mathrm{f}^{\prime}(\mathrm{x})=0\) \(2 a x+b=0\) \(x=-\frac{b}{2 a}\) At \(x=1 f^{\prime}(1)=2 a+b=0\) \(f(1)=a(1)^{2}+b(1)+4 \tag{i}\) \(-1=a+b+4\) \(a+b+5=0 \tag{ii}\) Equation (i) and (ii), we get \(a=5, \quad b=-10\)
BCECE-2016
Application of Derivatives
85672
The maximum value of \(12 \sin \theta-9 \sin ^{2} \theta\) is :
1 3
2 4
3 5
4 none of these
Explanation:
(B) : \(12 \sin \theta-9 \sin ^{2} \theta\) \(=4-4+12 \sin \theta-9 \sin ^{2} \theta=4-(2-3 \sin \theta)^{2}\) For maximum value, \((2-3 \sin \theta)^{2}\) gives the minimum value, \((2-3 \sin \theta)^{2} =0\) \(\sin \theta =\frac{2}{3}\) \(\therefore\) Maximum value, at \(\sin \theta=\frac{2}{3}\) \(12 \sin \theta-9 \sin ^{2} \theta\) \(=12 \times \frac{2}{3}-9 \times\left(\frac{2}{3}\right)^{2}=8-4=4\)
BCECE-2004
Application of Derivatives
85673
The function \(\frac{a \sin x+b \cos x}{c \sin x+d \cos x}, x \in R\) attains neither maximum nor minimum if,
85674
Which one of the following is correct in respect of the function \(f(x)=5 \log x-24 x+32 x^{2} ; x>0\) ?
1 It has neither a point of maximum nor a point of minimum
2 It has a point of maximum, but no point of minimum
3 It has a point of minimum, but no point of maximum.
4 It has a point of maximum and a point of minimum
Explanation:
(A) : Given, \(f(x)=5 \log x-24 x+32 x^{2} ; x>0\) Diff. w.r.t. \(\mathrm{x}\) we get- \(f^{\prime}(x)=\frac{5}{x}-24+64 x\) Put, \(\mathrm{f}^{\prime}(\mathrm{x})=0\) \(\frac{5}{x}-24+64 x=0\) \(\Rightarrow 64 x^{2}-24 x+5=0\) Now, \(D=(-24)^{2}-4 \times 5 \times 64\) \(D=576-1280\) \(D=-704\lt 0\) Now real value of \(x\) exist. Hence, it has neither a point of maxima nor a point of minima.
85671
If \(a x^{2}+b x+4\) attains its minimum value -1 at \(\mathbf{x}=1\), then the values of \(\mathbf{a}\) and \(b\) are respectively
1 \(5,-10\)
2 \(5,-5\)
3 5,5
4 \(10,-5\)
Explanation:
(A): \(f(x)=a x^{2}+b x+4\) \(f^{\prime}(x)=2 a x+b\) For local minima, \(\mathrm{f}^{\prime}(\mathrm{x})=0\) \(2 a x+b=0\) \(x=-\frac{b}{2 a}\) At \(x=1 f^{\prime}(1)=2 a+b=0\) \(f(1)=a(1)^{2}+b(1)+4 \tag{i}\) \(-1=a+b+4\) \(a+b+5=0 \tag{ii}\) Equation (i) and (ii), we get \(a=5, \quad b=-10\)
BCECE-2016
Application of Derivatives
85672
The maximum value of \(12 \sin \theta-9 \sin ^{2} \theta\) is :
1 3
2 4
3 5
4 none of these
Explanation:
(B) : \(12 \sin \theta-9 \sin ^{2} \theta\) \(=4-4+12 \sin \theta-9 \sin ^{2} \theta=4-(2-3 \sin \theta)^{2}\) For maximum value, \((2-3 \sin \theta)^{2}\) gives the minimum value, \((2-3 \sin \theta)^{2} =0\) \(\sin \theta =\frac{2}{3}\) \(\therefore\) Maximum value, at \(\sin \theta=\frac{2}{3}\) \(12 \sin \theta-9 \sin ^{2} \theta\) \(=12 \times \frac{2}{3}-9 \times\left(\frac{2}{3}\right)^{2}=8-4=4\)
BCECE-2004
Application of Derivatives
85673
The function \(\frac{a \sin x+b \cos x}{c \sin x+d \cos x}, x \in R\) attains neither maximum nor minimum if,
85674
Which one of the following is correct in respect of the function \(f(x)=5 \log x-24 x+32 x^{2} ; x>0\) ?
1 It has neither a point of maximum nor a point of minimum
2 It has a point of maximum, but no point of minimum
3 It has a point of minimum, but no point of maximum.
4 It has a point of maximum and a point of minimum
Explanation:
(A) : Given, \(f(x)=5 \log x-24 x+32 x^{2} ; x>0\) Diff. w.r.t. \(\mathrm{x}\) we get- \(f^{\prime}(x)=\frac{5}{x}-24+64 x\) Put, \(\mathrm{f}^{\prime}(\mathrm{x})=0\) \(\frac{5}{x}-24+64 x=0\) \(\Rightarrow 64 x^{2}-24 x+5=0\) Now, \(D=(-24)^{2}-4 \times 5 \times 64\) \(D=576-1280\) \(D=-704\lt 0\) Now real value of \(x\) exist. Hence, it has neither a point of maxima nor a point of minima.