Question

The function \(f(x) = 4 \sin^3 x - 6 \sin^2 x + 12 \sin x + 100\) is strictly

• A. decreasing in \(\left[ - \frac{\pi}{2}, \frac{\pi}{2} \right]\)
• B. increasing in \([ \pi, \frac{3\pi}{2} ]\)
• C. decreasing in \([0, \frac{\pi}{2}]\)
• D. decreasing in \([\frac{\pi}{2}, \pi]\)

Answer 1

The Correct Option is D

\(f(x) = 4 \sin^3(x) - 6 \sin^2(x) + 12 \sin(x) + 100\)
To find the derivative, we can use the chain rule and the power rule:
\(f'(x) = 3(4 \sin^2(x) \cos(x)) - 2(6 \sin(x) \cos(x)) + 12 \cos(x)\)
Simplifying further:
\(f'(x) = 12 \sin^2(x) \cos(x) - 12 \sin(x) \cos(x) + 12 \cos(x)\)

Now, to determine the intervals of increasing or decreasing, we need to analyze the sign of f'(x). 
Let's analyze the sign of f'(x) in different intervals: 
In the interval \([\frac{\pi}{2}, \pi]\)
For x values between \(\frac{\pi}{2}\) and \(π, sin(x)\) is positive, and cos(x) is negative. 
Since \(sin^2(x)\) and \(cos(x) \) are non-negative, while sin(x) is positive, all terms in f'(x) are positive. 
Therefore, f'(x) is always positive in this interval. 
Thus, f(x) is strictly increasing in the interval \([\frac{\pi}{2}, \pi]\)
Based on the analysis, we can conclude that the function 
\(f(x) = 4\sin^3(x) - 6\sin^2(x) + 12\sin(x) + 100\)  is strictly increasing in the interval \([\frac{\pi}{2}, \pi]\), which corresponds to option (D) decreasing in \([\frac{\pi}{2}, \pi].\)