Let f(x) = begincases x^2 sin (1/x ) & , x ≠ 0 \ 0 & , x = 0 endcases Then at x = 0

f is continuous but f' is not continuous

f' is continuous but not differentiable

f is continuous but not differentiable

Let f(x) = begincases x^2 sin (1/x ) & , x ≠ 0 \ 0 & , x =… — JEE Main 2023 Mathematics PYQ

JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)

Let

f(x) = \begin{cases} x^2 \sin\left(\dfrac{1}{x}\right) & , x \neq 0 \\ 0 & , x = 0 \end{cases}

Then at

x = 0

A.
$f'$ is continuous but not differentiable
B.
$f$ and $f'$ both are continuous
C.
$f$ is continuous but $f'$ is not continuous
D.
$f$ is continuous but not differentiable

Show answer and solution

Correct option: C

Solution

f is continuous at x=0 since x²sin(1/x) → 0 as x→0. f'(0) = lim[h→0] h²sin(1/h)/h = lim[h→0] h·sin(1/h) = 0, so f is also differentiable at x=0. For x≠0, f'(x) = 2x·sin(1/x) - cos(1/x). As x→0, the term cos(1/x) oscillates between -1 and 1 with no limit, so lim[x→0] f'(x) does not exist. Therefore f' is not continuous at x=0.

[JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)]

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