If f(x)=casesx², & if x≥ 1\\ x, & if x<1cases, then show that f is not differentiable at x = 1.

MathsCBSE Class 12 · Board exam 2023, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2023: question 22(a)

Set in the CBSE Class 12 board exam of 2023, paper 65/5/1, section B, question 22(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22(a) · 2 marks · Very short answerContinuity and Differentiability

If $f (x) = {x^{2}, x, if x \geq 1 if x < 1$ , then show that f is not differentiable at $x = 1$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/5/1.

Hints

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

Hint 1

Differentiability at $x = 1$ needs the left and right derivatives to be equal.

Hint 2

Use $f (1 + h) = (1 + h)^{2}$ on the right and $f (1 - h) = 1 - h$ on the left, with $f (1) = 1$ .

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.

Here $f (1) = 1^{2} = 1$ .
RHD: $lim_{h \to 0} \frac{f ( 1 + h ) - f ( 1 )}{h} = lim_{h \to 0} \frac{( 1 + h ) ^{2} - 1}{h} = lim_{h \to 0} \frac{2 h + h ^{2}}{h} = lim_{h \to 0} (2 + h) = 2$
1 mark
LHD: $lim_{h \to 0} \frac{f ( 1 - h ) - f ( 1 )}{- h} = lim_{h \to 0} \frac{( 1 - h ) - 1}{- h} = lim_{h \to 0} \frac{- h}{- h} = 1$
½ mark
Since LHD $= 1 \neq = 2 =$ RHD, $f$ is not differentiable at $x = 1$ .
½ mark

Answer: RHD = 2, LHD = 1; since they are unequal, f is not differentiable at x = 1.

Where the marks are

Correctly finds RHD = 2 and LHD = 1 (1½ marks)
Concludes non-differentiability because LHD ≠ RHD (½ 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 paper2023 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 22(a)

Hints

Step-by-step solution

Revise the chapter

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