Question

An extreme value of \( f(x) = \frac{4}{\sin x} + \frac{1}{1 - \sin x} \) in \( (0, \frac{\pi}{2}) \) is:

• A. \( 9 \)
• B. \( 8 \)
• C. \( \frac{2}{3} \)
• D. \( -\frac{7}{2} \)

Hint:

Convert trigonometric function to algebraic using substitution, then optimize.

Answer 1

The Correct Option is A

Let \( y = \sin x \), then: \[ f(y) = \frac{4}{y} + \frac{1}{1 - y}, \quad y \in (0, 1) \Rightarrow f'(y) = -\frac{4}{y^2} + \frac{1}{(1 - y)^2} \Rightarrow f'(y) = 0 \Rightarrow y = \frac{1}{3} \Rightarrow f = \frac{4}{1/3} + \frac{1}{1 - 1/3} = 12 + \frac{3}{2} = \boxed{9} \]