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

Answer: (D)

decreasing in $[\frac{\pi}{2}, \pi]$

Answer 2

$f(x) = 4 \sin^3(x) - 6 \sin^2(x) + 12 \sin(x) + 100$
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 $[\frac{\pi}{2}, \pi].$