Consider the function \(f(x) =\sin x\) in the interval \([\pi, 2\pi]\), then which of the following statements are correct?
(A) \(x =\displaystyle\frac{3\pi}{2}\) is its stationary point.
(B) Its maximum value is 1
(C) Its minimum value is −1
(D) It attains its maximum value at \(\pi\) and \(2\pi\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
On a closed interval, the absolute maximum and minimum are found by comparing f at the stationary points and at the end-points.
\(f'(x) = \cos x = 0\)
\(\displaystyle x = \frac{3\pi}{2}\) (the only solution in \([\pi, 2\pi]\))
\( \displaystyle f(\pi) = 0 \)
\( \displaystyle f\left(\frac{3\pi}{2}\right) = -1 \)
\( \displaystyle f(2\pi) = 0 \)
So the stationary point is \(x = \frac{3\pi}{2}\), the maximum value is 0 (at \(\pi\) and \(2\pi\)) and the minimum value is −1.
Checking the statements
(A) \(x =\frac{3\pi}{2}\) is its stationary point. — Correct. \(f'\left(\frac{3\pi}{2}\right) = \cos\frac{3\pi}{2} = 0\).
(B) Its maximum value is 1 — Incorrect. On \([\pi, 2\pi]\), \(\sin x \le 0\), so the maximum value is 0.
(C) Its minimum value is −1 — Correct. \(f\left(\frac{3\pi}{2}\right) = -1\) is the least value.
(D) It attains its maximum value at \(\pi\) and \(2\pi\) — Correct. \(f(\pi) = f(2\pi) = 0\), the maximum value.