# Math Help - Help on a challeging ABSOLUTE EXTREMA problem.

1. ## Help on a challeging ABSOLUTE EXTREMA problem.

I find this problem really though. I did solve for it's derivative and then by EVT tried to find its crititcal points. I get (-π/2) and (π/2) for f'(x)=0.
The problm is that I'm not sure if there are more solutions for f'(x).

Please see function in problem 1 of attachment:

Any help will be greatly apreciated.
Thank you.

2. Ok

you mean the first question right

the derivative of f(x) = 4x^3 + 12pie x cosx - 12 pie sinx + 3 pie^2 x

df/dx= 12 x^2 + 12pie cosx - 12 pie x sinx -12 pie cosx + 3 pie ^2

df/dx= 12x^2 - 12 pie x sinx + 3 pie^2

df/dx= 3 ( 4x^2 - 4pie x sinx + pie^2 )

( -b +- (b^2 - 4ac)^1/2 )/2

1/2(4 pie sinx +- (16pie^2 (sinx)^2 - 16 pie^2 )1/2 )

1/2(4 pie sinx +- (16pie^2 ((sinx)^2 - 1) )^1/2 )

1/2(4 pie sinx +- 4 pie( (sinx)^2 - 1 )1/2 )

1/2(4 pie sinx +- 4 pie( -(cosx)^2 )1/2 )

and -(cosx)^2 is a negative value for all values of x so you just have the value of x which make cosx zero like pie/2 + n pie n a natural number
the solution is pie /2 and -pie/2
there is no other solution

3. You'll eventually end up with the equation $\sin^2 x = 1$ which has the solution $x=\frac{\pi}{2}+n\pi,\ n \in Z$