If f:R→ R is defined by f(x)= x ³, show that f''(x) exists for all real x and find it.

f''(x)=cases6x,&x≥0\\-6x,&x<0cases=6 x , exists for all x R

MathsCBSE Class 12 · Sample paper, session 2024-25 · 5 marks · Long answer

CBSE Class 12 Mathematics sample paper 2024-25: question 34(a)

From the official CBSE sample question paper for 2024-25, section D, question 34(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(a) · 5 marks · Long answerContinuity and Differentiability

If $f : R \to R$ is defined by $f (x) = ∣ x ∣^{3}$ , show that $f^{''} (x)$ exists for all real $x$ and find it.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Remove the modulus by writing $f (x)$ separately for $x \geq 0$ and $x < 0$ .

Hint 2

The only doubtful point is $x = 0$ ; check left and right derivatives there, first for $f$ and then for $f^{'}$ .

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.

$f (x) = ∣ x ∣^{3} = {x^{3}, - x^{3}, x \geq 0 x < 0$ . For $x \neq = 0$ , $f$ is a polynomial on each side, hence differentiable. At $x = 0$ : LHD $= lim_{x \to 0^{-}} \frac{- x ^{3} - 0}{x} = lim_{x \to 0^{-}} (- x^{2}) = 0$ .
½ mark
RHD $= lim_{x \to 0^{+}} \frac{x ^{3} - 0}{x} = lim_{x \to 0^{+}} x^{2} = 0$ .
½ mark
LHD = RHD, so $f$ is differentiable at $x = 0$ with $f^{'} (0) = 0$ .
½ mark
Therefore $f^{'} (x) = {3 x^{2}, - 3 x^{2}, x \geq 0 x < 0$ .
1 mark
For $f^{'}$ at $x = 0$ : LHD $= lim_{x \to 0^{-}} \frac{- 3 x ^{2} - 0}{x} = lim_{x \to 0^{-}} (- 3 x) = 0$ .
½ mark
RHD $= lim_{x \to 0^{+}} \frac{3 x ^{2} - 0}{x} = lim_{x \to 0^{+}} 3 x = 0$ .
½ mark
LHD = RHD, so $f^{'}$ is differentiable at $0$ (and elsewhere, being polynomial on each side).
½ mark
Hence $f^{''} (x)$ exists for all real $x$ and $f^{''} (x) = {6 x, - 6 x, x \geq 0 x < 0$ , i.e. $f^{''} (x) = 6∣ x ∣$ .
1 mark

Answer: $f^{''} (x) = {6 x, - 6 x, x \geq 0 x < 0 = 6∣ x ∣$ , exists for all $x \in R$

Where the marks are

Writes $f$ piecewise and finds LHD of $f$ at 0 (½ mark)
RHD of $f$ at 0 (½ mark)
Concludes $f$ differentiable at 0 (½ mark)
Writes $f^{'} (x)$ piecewise (1 mark)
LHD of $f^{'}$ at 0 (½ mark)
RHD of $f^{'}$ at 0 (½ mark)
Concludes $f^{'}$ differentiable at 0 (½ mark)
States $f^{''} (x) = 6 x$ ( $x \geq 0$ ), $- 6 x$ ( $x < 0$ ) (1 mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)
Topic: Parametric and Second Order Derivatives

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 paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 34(a)

Hints

Step-by-step solution

Revise the chapter

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