<s>[INST] Extrema of $f(x)=\frac{\sin (5x)} 5 - \frac{2\sin(3x)} {3} + \sin (x)$. (a) I need help in finding maxima and minima of the following funcion:
$$f(x)=\frac{\sin (5x)} 5 - \frac{2\sin(3x)} {3} + \sin (x)$$
therefore I need to find the roots of $f'(x)=\cos(5x)-2\cos(3x)+\cos(x)$.
(b) I need to find the minima of $$f(x)=4x+\frac{9\pi^2} x + \sin x.$$
How can I find the roots of $$f'(x)=4-\frac{9\pi^2 }{x^2} + \cos x?$$
Thank you.
 [/INST] Hints:
a) $\cos(5x) \equiv 16\cos^5(x)-20\cos^3(x)+5\cos(x)$
$\cos(3x) \equiv 4\cos^3(x)-3\cos(x)$
So, we've got:
Now, let $c \equiv \cos(x)$.
We've got:
$$[16c^5-10c^3+5c]-2[4c^3-3c]+c=0 \iff \boxed{2c\underbrace{[8c^4-9c^2+6=0]}_{\textrm{quadratic in} \ c^2}} \tag{A}$$
Solve $(A)$!
b) Solving this analytically is very hard, so use the bisection method (noting that $4-\frac{9\pi^2}{x^2}=\cos(x)=0$ is continuous on $\mathbb{R \setminus\{0\}}$).
The exact answer to b) is $\boxed{\pm \frac{3\pi}{2}}$.
Now, your job is to evaluate the second derivative of these extrema, in order to determine their nature (maxima or minima).
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