MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2025: question 33

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerApplication of Derivatives

Find the absolute maximum and absolute minimum of function $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 1$ on $[1, 5]$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

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

Hint 1

On a closed interval, the absolute extrema occur at critical points or at end points.

Hint 2

Find where $f^{'} (x) = 0$ , then evaluate $f$ at those points and at $x = 1$ and $x = 5$ . Compare the values.

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.

Differentiate: $f^{'} (x) = 6 x^{2} - 30 x + 36 = 6 (x^{2} - 5 x + 6) = 6 (x - 2) (x - 3)$ .
1 mark
$f^{'} (x) = 0 \Rightarrow x = 2, 3$ . Both points lie in $[1, 5]$ .
1 mark
Evaluate $f$ at the critical points and the end points:
$f (1) = 2 - 15 + 36 + 1 = 24$
$f (2) = 16 - 60 + 72 + 1 = 29$
$f (3) = 54 - 135 + 108 + 1 = 28$
$f (5) = 250 - 375 + 180 + 1 = 56$
2 marks
Hence the absolute maximum value is 56 (at $x = 5$ ) and the absolute minimum value is 24 (at $x = 1$ ).
1 mark

Answer: Absolute maximum = 56 at x = 5; absolute minimum = 24 at x = 1

Where the marks are

Finds $f^{'} (x) = 6 (x - 2) (x - 3)$ (1 mark)
Gets critical points $x = 2, 3$ in $[1, 5]$ (1 mark)
Evaluates $f (1) = 24$ , $f (2) = 29$ , $f (3) = 28$ , $f (5) = 56$ (2 marks)
States absolute maximum 56 and absolute minimum 24 (1 mark)

Revise the chapter

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

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 paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 33

Hints

Step-by-step solution

Revise the chapter

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