80224
Let \([x]\) denote the greatest integer less than or equal to \(x\). Then the number of points where the function \(y=[x]+|1-x|,-1 \leq x \leq 3\) is not differentiable, is
1 1
2 2
3 3
4 4
Explanation:
(D) : Given, \(\mathrm{y}=[\mathrm{x}]+|1-\mathrm{x}|\) \(f(x)= \begin{cases}-1+1-x=-x ,-1\lt x \leq 0 \\ 0+1-x=1-x , 0 \leq x\lt 1 \\ x , 1 \leq x\lt 2 \\ 1+x , 2 \leq x\lt 3 \\ 5 , x=3\end{cases}\) The points are \(x=-1,0,1,2\) and 3 \(\mathrm{f}(\mathrm{x})\) is differentiable at \(\mathrm{x}=-1\) but non-differentiable at \(\mathrm{x}=0,1,2\) and 3 Hence, the number of points are 4.
AP EAMCET-2019-20.04.2019
Limits, Continuity and Differentiability
80225
If \(f(x)=\operatorname{Max}\{3-x, 3+x, 6\}\) is not differentiable at \(x=a\), and \(x=b\), then \(|\mathbf{a}|+|\mathbf{b}|=\)
1 4
2 5
3 6
4 8
Explanation:
(C) : Given, \(f(x)=\max (3-x, 3+x, 6)\) \(F(x)\) can be split based on the different value of \(x\). When, \(\mathrm{x}>3\) \(\mathrm{f}(\mathrm{x})=3+\mathrm{x}\) Let, \(\quad \mathrm{x}\lt -3\) \(\mathrm{f}(\mathrm{x})=3-\mathrm{x}\) Then, \(\quad-3 \leq \mathrm{x} \leq 3\) \(f(x)=6\)
AP EAMCET-2021-04.07.2021
Limits, Continuity and Differentiability
80226
If \(f(x)=|x|+|\sin x|\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), then its left hand derivative at \(x=0\) is
1 0
2 -1
3 -2
4 -3
Explanation:
(C) : Given, \(f(x)=|x|+|\sin x|\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) \(f(x)= \begin{cases}-x-\sin x \text { if }-\frac{\pi}{2}\lt x \leq 0 \\ x+\sin x \text { if } 0\lt x\lt \frac{\pi}{2}\end{cases}\) \(f\left(0^{-}\right)=\left.\frac{d}{d x}(-x-\sin x)\right|_{x=0}\) \(=-1-\cos x \mid=-1-1=-2\)
AP EAMCET-2011
Limits, Continuity and Differentiability
80227
If \(f: R \rightarrow R\) is an even function which is twice differentiable on \(R\) and \(f^{\prime \prime}(\pi)=1\), then \(f^{\prime \prime}(-\pi)\) is equal to
1 -1
2 0
3 1
4 2
Explanation:
(C) : Given, \(\mathrm{f} "(\pi)=1\) If \(f\) is an even function, \(\mathrm{f}^{\prime}(\mathrm{x})=\mathrm{f}(-\mathrm{x})\) \(\mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{f}^{\prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\mathrm{x})=-(-) \mathrm{f}^{\prime \prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\mathrm{x})=\mathrm{f}^{\prime \prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\pi)=\mathrm{f}^{\prime \prime}(-\pi)=1\)
NEET Test Series from KOTA - 10 Papers In MS WORD
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Limits, Continuity and Differentiability
80224
Let \([x]\) denote the greatest integer less than or equal to \(x\). Then the number of points where the function \(y=[x]+|1-x|,-1 \leq x \leq 3\) is not differentiable, is
1 1
2 2
3 3
4 4
Explanation:
(D) : Given, \(\mathrm{y}=[\mathrm{x}]+|1-\mathrm{x}|\) \(f(x)= \begin{cases}-1+1-x=-x ,-1\lt x \leq 0 \\ 0+1-x=1-x , 0 \leq x\lt 1 \\ x , 1 \leq x\lt 2 \\ 1+x , 2 \leq x\lt 3 \\ 5 , x=3\end{cases}\) The points are \(x=-1,0,1,2\) and 3 \(\mathrm{f}(\mathrm{x})\) is differentiable at \(\mathrm{x}=-1\) but non-differentiable at \(\mathrm{x}=0,1,2\) and 3 Hence, the number of points are 4.
AP EAMCET-2019-20.04.2019
Limits, Continuity and Differentiability
80225
If \(f(x)=\operatorname{Max}\{3-x, 3+x, 6\}\) is not differentiable at \(x=a\), and \(x=b\), then \(|\mathbf{a}|+|\mathbf{b}|=\)
1 4
2 5
3 6
4 8
Explanation:
(C) : Given, \(f(x)=\max (3-x, 3+x, 6)\) \(F(x)\) can be split based on the different value of \(x\). When, \(\mathrm{x}>3\) \(\mathrm{f}(\mathrm{x})=3+\mathrm{x}\) Let, \(\quad \mathrm{x}\lt -3\) \(\mathrm{f}(\mathrm{x})=3-\mathrm{x}\) Then, \(\quad-3 \leq \mathrm{x} \leq 3\) \(f(x)=6\)
AP EAMCET-2021-04.07.2021
Limits, Continuity and Differentiability
80226
If \(f(x)=|x|+|\sin x|\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), then its left hand derivative at \(x=0\) is
1 0
2 -1
3 -2
4 -3
Explanation:
(C) : Given, \(f(x)=|x|+|\sin x|\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) \(f(x)= \begin{cases}-x-\sin x \text { if }-\frac{\pi}{2}\lt x \leq 0 \\ x+\sin x \text { if } 0\lt x\lt \frac{\pi}{2}\end{cases}\) \(f\left(0^{-}\right)=\left.\frac{d}{d x}(-x-\sin x)\right|_{x=0}\) \(=-1-\cos x \mid=-1-1=-2\)
AP EAMCET-2011
Limits, Continuity and Differentiability
80227
If \(f: R \rightarrow R\) is an even function which is twice differentiable on \(R\) and \(f^{\prime \prime}(\pi)=1\), then \(f^{\prime \prime}(-\pi)\) is equal to
1 -1
2 0
3 1
4 2
Explanation:
(C) : Given, \(\mathrm{f} "(\pi)=1\) If \(f\) is an even function, \(\mathrm{f}^{\prime}(\mathrm{x})=\mathrm{f}(-\mathrm{x})\) \(\mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{f}^{\prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\mathrm{x})=-(-) \mathrm{f}^{\prime \prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\mathrm{x})=\mathrm{f}^{\prime \prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\pi)=\mathrm{f}^{\prime \prime}(-\pi)=1\)
80224
Let \([x]\) denote the greatest integer less than or equal to \(x\). Then the number of points where the function \(y=[x]+|1-x|,-1 \leq x \leq 3\) is not differentiable, is
1 1
2 2
3 3
4 4
Explanation:
(D) : Given, \(\mathrm{y}=[\mathrm{x}]+|1-\mathrm{x}|\) \(f(x)= \begin{cases}-1+1-x=-x ,-1\lt x \leq 0 \\ 0+1-x=1-x , 0 \leq x\lt 1 \\ x , 1 \leq x\lt 2 \\ 1+x , 2 \leq x\lt 3 \\ 5 , x=3\end{cases}\) The points are \(x=-1,0,1,2\) and 3 \(\mathrm{f}(\mathrm{x})\) is differentiable at \(\mathrm{x}=-1\) but non-differentiable at \(\mathrm{x}=0,1,2\) and 3 Hence, the number of points are 4.
AP EAMCET-2019-20.04.2019
Limits, Continuity and Differentiability
80225
If \(f(x)=\operatorname{Max}\{3-x, 3+x, 6\}\) is not differentiable at \(x=a\), and \(x=b\), then \(|\mathbf{a}|+|\mathbf{b}|=\)
1 4
2 5
3 6
4 8
Explanation:
(C) : Given, \(f(x)=\max (3-x, 3+x, 6)\) \(F(x)\) can be split based on the different value of \(x\). When, \(\mathrm{x}>3\) \(\mathrm{f}(\mathrm{x})=3+\mathrm{x}\) Let, \(\quad \mathrm{x}\lt -3\) \(\mathrm{f}(\mathrm{x})=3-\mathrm{x}\) Then, \(\quad-3 \leq \mathrm{x} \leq 3\) \(f(x)=6\)
AP EAMCET-2021-04.07.2021
Limits, Continuity and Differentiability
80226
If \(f(x)=|x|+|\sin x|\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), then its left hand derivative at \(x=0\) is
1 0
2 -1
3 -2
4 -3
Explanation:
(C) : Given, \(f(x)=|x|+|\sin x|\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) \(f(x)= \begin{cases}-x-\sin x \text { if }-\frac{\pi}{2}\lt x \leq 0 \\ x+\sin x \text { if } 0\lt x\lt \frac{\pi}{2}\end{cases}\) \(f\left(0^{-}\right)=\left.\frac{d}{d x}(-x-\sin x)\right|_{x=0}\) \(=-1-\cos x \mid=-1-1=-2\)
AP EAMCET-2011
Limits, Continuity and Differentiability
80227
If \(f: R \rightarrow R\) is an even function which is twice differentiable on \(R\) and \(f^{\prime \prime}(\pi)=1\), then \(f^{\prime \prime}(-\pi)\) is equal to
1 -1
2 0
3 1
4 2
Explanation:
(C) : Given, \(\mathrm{f} "(\pi)=1\) If \(f\) is an even function, \(\mathrm{f}^{\prime}(\mathrm{x})=\mathrm{f}(-\mathrm{x})\) \(\mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{f}^{\prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\mathrm{x})=-(-) \mathrm{f}^{\prime \prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\mathrm{x})=\mathrm{f}^{\prime \prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\pi)=\mathrm{f}^{\prime \prime}(-\pi)=1\)
80224
Let \([x]\) denote the greatest integer less than or equal to \(x\). Then the number of points where the function \(y=[x]+|1-x|,-1 \leq x \leq 3\) is not differentiable, is
1 1
2 2
3 3
4 4
Explanation:
(D) : Given, \(\mathrm{y}=[\mathrm{x}]+|1-\mathrm{x}|\) \(f(x)= \begin{cases}-1+1-x=-x ,-1\lt x \leq 0 \\ 0+1-x=1-x , 0 \leq x\lt 1 \\ x , 1 \leq x\lt 2 \\ 1+x , 2 \leq x\lt 3 \\ 5 , x=3\end{cases}\) The points are \(x=-1,0,1,2\) and 3 \(\mathrm{f}(\mathrm{x})\) is differentiable at \(\mathrm{x}=-1\) but non-differentiable at \(\mathrm{x}=0,1,2\) and 3 Hence, the number of points are 4.
AP EAMCET-2019-20.04.2019
Limits, Continuity and Differentiability
80225
If \(f(x)=\operatorname{Max}\{3-x, 3+x, 6\}\) is not differentiable at \(x=a\), and \(x=b\), then \(|\mathbf{a}|+|\mathbf{b}|=\)
1 4
2 5
3 6
4 8
Explanation:
(C) : Given, \(f(x)=\max (3-x, 3+x, 6)\) \(F(x)\) can be split based on the different value of \(x\). When, \(\mathrm{x}>3\) \(\mathrm{f}(\mathrm{x})=3+\mathrm{x}\) Let, \(\quad \mathrm{x}\lt -3\) \(\mathrm{f}(\mathrm{x})=3-\mathrm{x}\) Then, \(\quad-3 \leq \mathrm{x} \leq 3\) \(f(x)=6\)
AP EAMCET-2021-04.07.2021
Limits, Continuity and Differentiability
80226
If \(f(x)=|x|+|\sin x|\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), then its left hand derivative at \(x=0\) is
1 0
2 -1
3 -2
4 -3
Explanation:
(C) : Given, \(f(x)=|x|+|\sin x|\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) \(f(x)= \begin{cases}-x-\sin x \text { if }-\frac{\pi}{2}\lt x \leq 0 \\ x+\sin x \text { if } 0\lt x\lt \frac{\pi}{2}\end{cases}\) \(f\left(0^{-}\right)=\left.\frac{d}{d x}(-x-\sin x)\right|_{x=0}\) \(=-1-\cos x \mid=-1-1=-2\)
AP EAMCET-2011
Limits, Continuity and Differentiability
80227
If \(f: R \rightarrow R\) is an even function which is twice differentiable on \(R\) and \(f^{\prime \prime}(\pi)=1\), then \(f^{\prime \prime}(-\pi)\) is equal to
1 -1
2 0
3 1
4 2
Explanation:
(C) : Given, \(\mathrm{f} "(\pi)=1\) If \(f\) is an even function, \(\mathrm{f}^{\prime}(\mathrm{x})=\mathrm{f}(-\mathrm{x})\) \(\mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{f}^{\prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\mathrm{x})=-(-) \mathrm{f}^{\prime \prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\mathrm{x})=\mathrm{f}^{\prime \prime}(-\mathrm{x})\) \(\mathrm{f}^{\prime \prime}(\pi)=\mathrm{f}^{\prime \prime}(-\pi)=1\)