Consider \(f(x)=\sin(3x) + 4, \forall x \in R\), then
(A) Maximum value of \(f(x)\) is 5
(B) Minimum value of \(f(x)\) is 3
(C) Maximum value of \(f(x)\) is attained at \(x=\displaystyle\frac{\pi}{6}\)
(D) Minimum value of \(f(x)\) is attained at \(x = 0\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
The range of sine is \([-1, 1]\) for every angle, so
\(-1 \le \sin 3x \le 1\)
\(3 \le \sin 3x + 4 \le 5\)
The maximum 5 occurs when \(\sin 3x = 1\), for example at \(3x = \frac{\pi}{2}\), that is \(x = \frac{\pi}{6}\).
The minimum 3 occurs when \(\sin 3x = -1\), for example at \(3x = \frac{3\pi}{2}\), that is \(x = \frac{\pi}{2}\).
So the maximum value is 5, attained at \(x = \frac{\pi}{6}\); the minimum value is 3; and \(f(0) = 4\).
Checking the statements
(A) Maximum value of \(f(x)\) is 5 — Correct. \(1 + 4 = 5\).
(B) Minimum value of \(f(x)\) is 3 — Correct. \(-1 + 4 = 3\).
(C) Maximum value of \(f(x)\) is attained at \(x=\frac{\pi}{6}\) — Correct. \(f\left(\frac{\pi}{6}\right) = \sin\frac{\pi}{2} + 4 = 5\).
(D) Minimum value of \(f(x)\) is attained at \(x = 0\) — Incorrect. \(f(0) = \sin 0 + 4 = 4\), not 3.