Check whether the function f(x) = x² x is differentiable at x = 0 or not.

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2024: question 21(a)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section B, question 21(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21(a) · 2 marks · Very short answerContinuity and Differentiability

Check whether the function $f (x) = x^{2} ∣ x ∣$ is differentiable at $x = 0$ or not.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Write $∣ x ∣$ as $x$ for $x \geq 0$ and $- x$ for $x < 0$ , then express $f (x)$ in two pieces.

Hint 2

Compute the left-hand and right-hand derivatives at 0 using the limit definition with $h \to 0$ .

Hint 3

Compare the two limits to decide differentiability.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Remove the modulus: $f (x) = {x^{3}, - x^{3}, x \geq 0 x < 0$ , so $f (0) = 0$ .
½ mark
RHD at $0$ : $lim_{h \to 0} \frac{f ( 0 + h ) - f ( 0 )}{h} = lim_{h \to 0} \frac{h ^{3} - 0}{h} = lim_{h \to 0} h^{2} = 0$
½ mark
LHD at $0$ : $lim_{h \to 0} \frac{f ( 0 - h ) - f ( 0 )}{- h} = lim_{h \to 0} \frac{- ( - h ) ^{3} - 0}{- h} = lim_{h \to 0} \frac{h ^{3}}{- h} = lim_{h \to 0} (- h^{2}) = 0$
½ mark
Since LHD = RHD = 0, $f (x)$ is differentiable at $x = 0$ , with $f^{'} (0) = 0$ .
½ mark

Answer: f is differentiable at x = 0 (f'(0) = 0)

Where the marks are

Writes f(x) piecewise without modulus (0.5 mark)
Finds RHD = 0 (0.5 mark)
Finds LHD = 0 (0.5 mark)
Concludes differentiable since LHD = RHD (0.5 mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Continuity and Differentiability, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 21(a)

Hints

Step-by-step solution

Revise the chapter

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