For the function f(x) = ex x ; x R, which of the following statements is/are TRUE?

The function is differentiable at all $x$

GATE 2023 CE (CE1) – Question 48

Engineering Mathematics · Calculus: Limits, continuity, mean value theorems, maxima and minima, Taylor series · 2 marks · Multiple select

For the function $f(x) = e^x|\sin x|$; $x \in \mathbb{R}$, which of the following statements is/are TRUE?

The function is continuous at all $x$
The function is differentiable at all $x$
The function is periodic
The function is bounded

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Show answer and explanation

Correct answer: (A) The function is continuous at all $x$

Explanation

The product of continuous functions is continuous (A true). At $x = n\pi$ the function $|\sin x|$ has a corner (the left and right slopes are $\mp 1$), and $e^x \neq 0$, so $f$ is not differentiable there (B false). The factor $e^x$ increases without limit, so $f$ is neither periodic nor bounded (C and D false).