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If $f(x)=\left\{\begin{array}{cc}-x-\frac{\pi}{2}, & x \leq-\frac{\pi}{2} \\ -\cos x, & -\frac{\pi}{2} < x \leq 0 \\ x-1, & 0 < x \leq 1 \\ \log x, & x>1\end{array}\right.$, then
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Verified Answer
The correct answers are:
$f(x)$ is continuous at $x=-\frac{\pi}{2}$
,
$f(x)$ is not differentiable at $x=0$
,
$f(x)$ is differentiable at $x=1$
,
$f(x)$ is differentiable at $x=-\frac{3}{2}$
$f(x)$ is continuous at $x=-\frac{\pi}{2}$
,
$f(x)$ is not differentiable at $x=0$
,
$f(x)$ is differentiable at $x=1$
,
$f(x)$ is differentiable at $x=-\frac{3}{2}$
$f(x)=\left\{\begin{array}{cc}-x-\frac{\pi}{2}, & x \leq-\frac{\pi}{2} \\ -\cos x, & -\frac{\pi}{2} < x \leq 0 \\ x-1, & 0 < x \leq 1 \\ \log x, & x>1\end{array}\right.$
Continuity at $x=-\frac{\pi}{2}$,
$$
\begin{aligned}
& f\left(-\frac{\pi}{2}\right)=-\left(-\frac{\pi}{2}\right)-\frac{\pi}{2}=0 \\
& \mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0}-\cos \left(-\frac{\pi}{2}+h\right)=0
\end{aligned}
$$
$\therefore$ Continuous at $x=0$
Continuity at $x=0 \Rightarrow f(0)=-1$ $\mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0}(0+h)-1=-1$
$\therefore$ Continuous at $x=0$
Continuity at $x=1 ; f(1)=0$ $\mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0} \log (1+h)=0$
$\therefore$ Continuous at $x=1$
Here, $f^{\prime}(x)=\left\{\begin{array}{cc}-1, & x \leq-\frac{\pi}{2} \\ \sin x, & -\frac{\pi}{2} < x \leq 0 \\ 1, & 0 < x \leq 1 \\ \frac{1}{x}, & x>1\end{array}\right.$
Differentiable at $x=0$, $\mathrm{LHD}=0, \mathrm{RHD}=1$
$\therefore \quad$ Not differentiable at $x=0$
Differentiable at $x=1$, $\mathrm{LHD}=1, \mathrm{RHD}=1$
$\therefore$ Differentiable at $x=1$
Also, for $x=-\frac{3}{2} \Rightarrow f(x)=-x-\frac{3}{2}$
$\therefore$ Differentiable at $x=-\frac{3}{2}$
Continuity at $x=-\frac{\pi}{2}$,
$$
\begin{aligned}
& f\left(-\frac{\pi}{2}\right)=-\left(-\frac{\pi}{2}\right)-\frac{\pi}{2}=0 \\
& \mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0}-\cos \left(-\frac{\pi}{2}+h\right)=0
\end{aligned}
$$
$\therefore$ Continuous at $x=0$
Continuity at $x=0 \Rightarrow f(0)=-1$ $\mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0}(0+h)-1=-1$
$\therefore$ Continuous at $x=0$
Continuity at $x=1 ; f(1)=0$ $\mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0} \log (1+h)=0$
$\therefore$ Continuous at $x=1$
Here, $f^{\prime}(x)=\left\{\begin{array}{cc}-1, & x \leq-\frac{\pi}{2} \\ \sin x, & -\frac{\pi}{2} < x \leq 0 \\ 1, & 0 < x \leq 1 \\ \frac{1}{x}, & x>1\end{array}\right.$
Differentiable at $x=0$, $\mathrm{LHD}=0, \mathrm{RHD}=1$
$\therefore \quad$ Not differentiable at $x=0$
Differentiable at $x=1$, $\mathrm{LHD}=1, \mathrm{RHD}=1$
$\therefore$ Differentiable at $x=1$
Also, for $x=-\frac{3}{2} \Rightarrow f(x)=-x-\frac{3}{2}$
$\therefore$ Differentiable at $x=-\frac{3}{2}$
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