CUET PG 2023 — Mathematics PYQ

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Question

CUET PG 2023 — Mathematics PYQ

CUET PG | Mathematics | 2023

Which of the following functions is differentiable at x = 0?

Choose the correct answer:

Explanation

Differentiability of a Function: A function $f(x)$ is differentiable at $x = a$ in its domain if its derivative is continuous at $a$ .

This means that $f'(a)$ must exist, or equivalently:

\lim_{x \to a^+} f'(x) = \lim_{x \to a^-} f'(x) = \lim_{x \to a} f'(x) = f'(a) \text{}

The Modulus Function $|x|$ is defined as:

|x| = \begin{cases} x, & x > 0 \\ -x, & x < 0 \\ 0, & x = 0 \end{cases} \text{}

Trigonometric Functions for Negative Angle:

\sin(-x) = -\sin x, \quad \cos(-x) = \cos x, \quad \tan(-x) = -\tan x \text{}

Derivatives of Trigonometric Functions:

$\frac{d}{dx} \sin x = \cos x \text{}$
$\frac{d}{dx} \cos x = -\sin x \text{}$
$\frac{d}{dx} \tan x = \sec^2 x \text{}$
$\frac{d}{dx} \cot x = -\csc^2 x \text{}$
$\frac{d}{dx} \sec x = \tan x \sec x \text{}$
$\frac{d}{dx} \csc x = -\cot x \csc x \text{}$

Solution: Since we have to find the function which is differentiable at $x = 0$ , let us find the values of $f'(0)$ for all the options.

As we can see from the above table that both the right limit $x \to 0^+$ and the left limit $x \to 0^-$ of $f'(x)$ is same only for $\sin(|x|) - |x|$ , so it is differentiable at $x = 0$ .