Let \(f(x) = \sin(x) \) and \(g(x) = x^3 \) be functions defined on \( [0, \frac{\pi}{2}] \) .
Assume that f(x) and g(x) satisfy the conditions of the Cauchy Mean Value Theorem.
Then, the value of \(c \in (0, \frac{\pi}{2}) \) such that
\(\frac{f'(c)}{g'(c)} = \frac{f(\frac{\pi}{2}) - f(0)}{g(\frac{\pi}{2}) - g(0)} \)
is given by solving which of the following equations?
1
\( \cos(c) = \frac{6c^2}{\pi^3} \)
2
\( \cos(c) = \frac{2c^2}{\pi^3} \)
3
\(\cos(c) = \frac{24c^2}{\pi^3} \)
4
\( \cos(c) = \frac{c^2}{\pi^3} \)