Show that the function f(x) = 4x³ - 18x² + 27x - 7 has neither maxima nor minima.

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

CBSE Class 12 Mathematics 2024: question 22

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, 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

Show that the function $f (x) = 4 x^{3} - 18 x^{2} + 27 x - 7$ has neither maxima nor minima.

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

Find $f^{'} (x)$ and look for a common factor.

Hint 2

Check whether $4 x^{2} - 12 x + 9$ is a perfect square.

Hint 3

If $f^{'}$ never changes sign, can there be a turning point?

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.

$f^{'} (x) = 12 x^{2} - 36 x + 27$
½ mark
$f^{'} (x) = 3 (4 x^{2} - 12 x + 9) = 3 (2 x - 3)^{2} \geq 0$ for all $x \in R$
½ mark
$f^{'} (x) = 0$ only at $x = \frac{3}{2}$ and $f^{'}$ does not change sign there, so $f$ is increasing on $R$ .
½ mark
An increasing function has no point where $f^{'}$ changes sign, hence $f (x)$ has neither a maximum nor a minimum.
½ mark

Answer: $f^{'} (x) = 3 (2 x - 3)^{2} \geq 0$ , so f is increasing on R and has neither maxima nor minima.

Where the marks are

Finds $f^{'} (x) = 12 x^{2} - 36 x + 27$ (0.5 mark)
Writes it as $3 (2 x - 3)^{2} \geq 0$ (0.5 mark)
States f is increasing on R (0.5 mark)
Concludes no maxima/minima (0.5 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 paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 22

Hints

Step-by-step solution

Revise the chapter

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