Question
For the function $f(x) = e^{sin|x|} - |x|,x \in \mathbf{R}$, consider the following statements:
Statement I : f is differentiable for all $x \in \mathbf{R}$.
Statement II : f is increasing in $\left( - \pi, - \frac{\pi}{2} \right)$.
In the light of the above statements, choose the correct answer from the options given below :
Options
Solution
Sol. [Key Concept: $f(x) = e^{sin|x|} - |x|$ is non-differentiable at $x = \pi$ (Statement I false), and on $( - \pi, - \frac{\pi}{2)}$ its derivative is positive so it increases (Statement II true).]
$f(x) = e^{|sinx|} - |x|$ is non-differentiable at $x = \pi$
and for $x \in \left( - \pi,\frac{- \pi}{2} \right)$
$${f(x) = e^{- sinx} + x }{f^{'}(x) = - e^{- sinx}cosx + 1 }$$for $x \in \left( - \pi,\frac{- \pi}{2} \right)\ cosx < 0$
$$\therefore f^{'}(x) > 0 $$$\therefore f(x)$ is increasing
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