Finding minimum values of a function

Question

I am trying to find the minimum values for the following function: $f(x)=(x-3)^4 + (x-5)^4 + (x-9)^4 + (x+10)^4$

Any hint is greatly appreciated.

Answer 1

$$f(x)=(x-3)^4 + (x-5)^4 + (x-9)^4 + (x+10)^4$$ $$f'(x)=16 x^3-84 x^2+2580 x+476=0$$

If you follow the steps given here for one real solution, $f'(x)=0$ when $$x_*=\frac{7}{4}-\frac{\sqrt{811}}{2} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{9639}{811 \sqrt{811}}\right)\right)\approx -0.183363$$ Since this number is much smaller than $3$, $5$, $9$ and $10$ you have a good approximation computing $f(0)=17267$ instead of $f(x_*)=17223.3$.