Check whether the function f: R→ R defined by f(x)=x³+x, has any critical point/s or not ? If yes, then find the point/s.

MathsCBSE Class 12 · Sample paper, session 2023-24 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2023-24: question 25

From the official CBSE sample question paper for 2023-24, section B, question 25, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q25 · 2 marks · Very short answerApplication of Derivatives

Check whether the function $f : R \to R$ defined by $f (x) = x^{3} + x$ , has any critical point/s or not ? If yes, then find the point/s.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

A critical point is where $f^{'} (x) = 0$ or $f^{'}$ does not exist.

Hint 2

Can $3 x^{2} + 1$ ever be zero for real $x$ ?

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.

$f^{'} (x) = 3 x^{2} + 1$ . Since $x^{2} \geq 0$ for all $x \in R$ , $f^{'} (x) \geq 1 > 0$ , so $f^{'} (x) \neq = 0$ for every real $x$ . Also $f$ is a polynomial, so $f^{'}$ exists everywhere.
1½ marks
Hence $f$ has no critical point.
½ mark

Answer: No critical point exists.

Where the marks are

Finds $f^{'} (x) = 3 x^{2} + 1 > 0$ for all real $x$ (1½ marks)
Concludes that there is no critical point (½ mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 25

Hints

Step-by-step solution

Revise the chapter

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