Choose the correct answers from the given four options : The func… - Vidyadip

MCQ

Choose the correct answers from the given four options : The function $\text{f(x)}=\text{e}^{|\text{x}|}$ is :

✓
Continuous everywhere but not differentiable at $x = 0.$
B
Continuous and differentiable everywhere.
C
Not continuous at $x = 0.$
D
None of these.

✓

Answer

Correct option: A.

Continuous everywhere but not differentiable at $x = 0.$

Let $\text{u(x)}=|\text{x}|$ and $\text{v(x)}=\text{e}^\text{x}$
$\therefore\ \text{f(x)}=\text{vou(x)}=\text{v}[\text{u(x)]}$
$=\text{v}|\text{x}|=\text{e}^{|\text{x}|}$
Since, $u(x)$ and $v(x)$ are both continuous functions.
So, $f(x)$ is also continuous function but $u(x) = |x|$ is not differentiable at $x = 0$, whereas $v(x) = e^x$ is differentiable at everywhere.
Hence, $f(x)$ is continuous everywhere but not differentiable at $x = 0.$

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