Continuity and Differentiability 4 marks Questions, CBSE Class 12 PYQs 2026

1 different 4 marks questions on Continuity and Differentiability from CBSE Class 12 Maths board exams 2026, newest first.

Q384 marksCase StudyContinuity and Differentiability CBSE 2026 · 65/2/1

Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes.
The equation of one such track is given as follows :
$f (x) = {x^{4} - 4 x^{2} + 4, x^{2} + 40, 0 \leq x < 3 x \geq 3$
Based on given information, answer the following questions :
(i) Find $f^{'} (x)$ for $0 < x < 3$ . (1)
(ii) Find $f^{'} (4)$ (1)
(iii) (a) Test for continuity of $f (x)$ at $x = 3$ . (2)
OR (iii) (b) Test for differentiability of $f (x)$ at $x = 3$ . (2)

Show answer & solution

Answer: (i)

4 x^{3} - 8 x

(ii) 8 (iii)(a) continuous at

x = 3

OR (iii)(b) not differentiable at

x = 3

(i) $f^{'} (x) = 4 x^{3} - 8 x$ for $0 < x < 3$ .
(ii) For $x > 3$ , $f^{'} (x) = 2 x$ , so $f^{'} (4) = 8$ .
(iii)(a) $lim_{x \to 3^{-}} f (x) = 81 - 36 + 4 = 49$ , $lim_{x \to 3^{+}} f (x) = 9 + 40 = 49$ , $f (3) = 49$ ; so f is continuous at $x = 3$ .
(iii)(b) LHD $= lim_{h \to 0^{+}} \frac{f ( 3 - h ) - f ( 3 )}{- h} = 4 (27) - 8 (3) = 84$ ; RHD $= lim_{h \to 0^{+}} \frac{f ( 3 + h ) - f ( 3 )}{h} = lim_{h \to 0^{+}} \frac{6 h + h ^{2}}{h} = 6$ .
LHD $\neq =$ RHD, so f is not differentiable at $x = 3$ .

Also asked in: 2026 65/2/2, 2026 65/2/3

Continuity and Differentiability: 4 marks Questions (CBSE Class 12)