Show that the function f(x) = (16 x)/(4 + x) - x, is strictly decreasing in ((π)/(2), π ).

f'(x)=( x(56- x))/((4+ x)²)<0 in (π/2,π), so f is strictly decreasing there.

MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2023: question 25

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, 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

Show that the function $f (x) = \frac{16 s i n x}{4 + c o s x} - x$ , is strictly decreasing in $(\frac{π}{2}, π)$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

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

Hint 1

Differentiate using the quotient rule and use $sin^{2} x + cos^{2} x = 1$ .

Hint 2

Factor the numerator. Then look at the sign of $cos x$ in the second quadrant.

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.

Using the quotient rule,
$f^{'} (x) = \frac{16 c o s x ( 4 + c o s x ) + 16 s i n ^{2} x}{( 4 + c o s x ) ^{2}} - 1$
1 mark
The numerator becomes $64 cos x + 16 cos^{2} x + 16 sin^{2} x = 64 cos x + 16$ . Hence
$f^{'} (x) = \frac{64 c o s x + 16 - ( 16 + 8 c o s x + c o s ^{2} x )}{( 4 + c o s x ) ^{2}} = \frac{c o s x ( 56 - c o s x )}{( 4 + c o s x ) ^{2}}$
½ mark
In $(\frac{π}{2}, π)$ we have $cos x < 0$ , while $56 - cos x > 0$ and $(4 + cos x)^{2} > 0$ . So $f^{'} (x) < 0$ .
Hence $f$ is strictly decreasing in $(\frac{π}{2}, π)$ .
½ mark

Answer: $f^{'} (x) = \frac{c o s x ( 56 - c o s x )}{( 4 + c o s x ) ^{2}} < 0$ in $(π /2, π)$ , so $f$ is strictly decreasing there.

Where the marks are

Correct $f^{'} (x)$ using the quotient rule (1 mark)
Simplifies to $\frac{c o s x ( 56 - c o s x )}{( 4 + c o s x ) ^{2}}$ (½ mark)
Sign argument with $cos x < 0$ and conclusion (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 paper2023 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 25

Hints

Step-by-step solution

Revise the chapter

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