Calculus

Continuity & Differentiability

Medium

core

Let $f(x) = |3-x| + (3+x)$ be a function, then which of the following statements are TRUE?

A. $f(x)$ is continuous at $x=3$

B. $f(x)$ is continuous at $x=-3$

C. $f(x)$ is differentiable at $x=3$

D. $f(x)$ is differentiable at $x=-3$

Choose the correct answer from the options given below:

A, B and D only

A, B, C and D

A, and B only

A, B and C only

✅ Correct Option: 1

The modulus function is continuous everywhere, so $f$ is continuous at both $x=3$ and $x=-3$ .

The corner of $|3-x|$ occurs at $x=3$ , where the left derivative is $0$ and the right derivative is $2$ , so $f$ is not differentiable there.

At $x=-3$ the function is smooth, so it is differentiable. Hence A, B and D.

2026: 29 May Shift 2

If $x=4t^3$ and $y=6t^2$ then $\frac{d^2y}{dx^2}$ at $t=1$ is

2022: 16 July Shift 2

Match List I with List II

List I	List II
A. If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$ , then $\frac{g(x)}{f(x)}$ is	I. discontinuous at exactly three points.
B. The function $f(x) = \frac{4-x^2}{4x-x^3}$ is	II. continuous everywhere
C. The function $f(x) = \lvert x \rvert + \lvert x-1 \rvert$ is	III. discontinuous at $x = 0$ .
D. The function $f(x) = \lvert \sin x \rvert$ is	IV. continuous at $x = 0$ and $x = 1$

Choose the correct answer from the options given below:

2025: 3 June Shift 1

Match List-I with List-II

List-I	List-II
Function	Points of discontinuity
(A) $f(x) = \frac{x^2 + 1}{x}$	(I) $x = 4$
(B) $f(x) = \frac{\vert x - 1 \vert}{x - 1}$	(II) $x = 2$
(C) $f(x) = \begin{cases} x - 1, & x < 2 \\ x + 1, & x \ge 2 \end{cases}$	(III) $x = 0$
(D) $f(x) = \frac{1 - x}{(x - 4)}$	(IV) $x = 1$

Choose the correct answer from the options given below:

2022: 7 Aug Shift 2

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$ , then the value of k is :

2025: 13 May Shift 1

If $y = 3^x + e^x + x^x + x^3$ , then the value of $\frac{dy}{dx}$ at $x = 3$ is

2026: 22 May Shift 1

Let a function $f(x)=\alpha+(\beta^2+5\beta+6)|x|+\gamma|x|^4$ , where $\alpha, \beta$ and $\gamma$ are real number. Then the function $f(x)$ is differentiable at $x=0$ if

Solution

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