For the function f(x)=|sin 4x+3| , then which of the following statem…

The correct answer is Option (2) → (A) and (C) only

By the bounds of the sine function,

    \( \displaystyle -1\le\sin 4x\le 1 \)

    \( \displaystyle 2\le\sin 4x+3\le 4 \)

Since \( \sin 4x+3 \) is always positive, the modulus changes nothing, so \( f(x)=\sin 4x+3 \).

The value 2 is attained when \( \sin 4x=-1 \) (for example \( x=\frac{3\pi}{8} \)), and the value 4 when \( \sin 4x=1 \) (for example \( x=\frac{\pi}{8} \)).

So the minimum value of f(x) is 2 and the maximum value is 4.

Checking the statements

(A) Minimum value of \( f(x) \) is 2 — Correct. It is attained when \( \sin 4x=-1 \).

(B) There is no minimum value — Incorrect. The minimum value 2 is attained.

(C) Maximum value of \( f(x) \) is 4 — Correct. It is attained when \( \sin 4x=1 \).

(D) Maximum value of \( f(x) \) is 3 — Incorrect. 3 is the value when \( \sin 4x=0 \); the maximum is 4.