Find the maximum profit that a company can make, if the profit function is given by P(x)=72+42x-x², where x is the number of units and P is the profit in rupees.

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

CBSE Class 12 Mathematics sample paper 2023-24: question 23(b)

From the official CBSE sample question paper for 2023-24, section B, question 23(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(b) · 2 marks · Very short answerCompetency-basedApplication of Derivatives

Find the maximum profit that a company can make, if the profit function is given by $P (x) = 72 + 42 x - x^{2}$ , where $x$ is the number of units and $P$ is the profit in rupees.

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 $P (x)$ and equate the derivative to zero.

Hint 2

Check the second derivative, then put the critical value back into $P (x)$ .

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.

$P^{'} (x) = 42 - 2 x$ . For a maximum or minimum, $P^{'} (x) = 0 \Rightarrow x = 21$ .
½ mark
$P^{''} (x) = - 2 < 0$ , so $P$ is maximum at $x = 21$ .
½ mark
Maximum profit $= P (21) = 72 + 42 (21) - (21)^{2} = 72 + 882 - 441 = ₹513$ .
1 mark

Answer: Maximum profit = ₹513 (at $x = 21$ units)

Where the marks are

Sets $P^{'} (x) = 0$ and gets $x = 21$ (½ mark)
Shows $P^{''} (21) < 0$ , so a maximum (½ mark)
Computes the maximum profit ₹513 (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 paperSample paper 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 23(b)

Hints

Step-by-step solution

Revise the chapter

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