For the function , , consider the following statements: Statement I: f is differentiable for all . Statement II: f is increasing in . In the light of the above statements, choose the correct answer from the options given below: [2026]

Question

For the function $f (x) = e^{\sin | x |} - | x |$ , $x \in R$ , consider the following statements:

Statement I: f is differentiable for all $x \in R$ .

Statement II: f is increasing in $(- π, - \frac{π}{2})$ .

In the light of the above statements, choose the correct answer from the options given below: [2026]

Smart Achievers · Accepted Answer

(1)

$f (x) = e^{\sin | x |} - | x |$ , $x \in R$

For Statement I

For $x \geq 0, f (x) = e^{\sin x} - x$

RHD at x = 0

$f' (0^{+}) = e^{0} . 1 - 1 = 0$

For $x < 0, f (x) = e^{- \sin x} + x$

LHD at x = 0

$f' (0^{-}) = - e^{0} . 1 + 1 = 0$

LHD = RHD at x = 0

$∴$ Statement I is true

For Statement II

$f' (x) = 1 - e^{- \sin x} . \cos x$ $[∵ x \in (- π, \frac{- π}{2})]$

Since, $- e^{- \sin x} . \cos x$ is +ve for given x.

So, $f' (x) = 1 + (a' + ve' value)$

$f' (x) > 0$ for $(- π, - \frac{π}{2})$

So Statement II is true.

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