Find the interval/s in which the function f: R→ R defined by f(x)=xe^x, is increasing.

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

CBSE Class 12 Mathematics sample paper 2023-24: question 22

From the official CBSE sample question paper for 2023-24, section B, question 22, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22 · 2 marks · Very short answerApplication of Derivatives

Find the interval/s in which the function $f : R \to R$ defined by $f (x) = x e^{x}$ , is increasing.

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

Differentiate $x e^{x}$ using the product rule.

Hint 2

The factor $e^{x}$ is always positive, so only the sign of $(x + 1)$ matters.

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.

By the product rule, $f^{'} (x) = x e^{x} + e^{x} = e^{x} (x + 1)$ .
1 mark
Since $e^{x} > 0$ for all $x \in R$ , $f^{'} (x) \geq 0 ⟺ x + 1 \geq 0 ⟺ x \geq - 1$ . Hence $f$ is increasing on $[- 1, \infty)$ .
1 mark

Answer: $[- 1, \infty)$

Where the marks are

Finds $f^{'} (x) = e^{x} (x + 1)$ (1 mark)
Uses $e^{x} > 0$ to conclude $f$ is increasing on $[- 1, \infty)$ (1 mark)

Revise the chapter

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

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 22

Hints

Step-by-step solution

Revise the chapter

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