I need some help with the following problem. Find the maximum value of the function $f(x)=|3x^2+2ax-1|$ for $x\in[-1,1]$ if $-2 \leq a \leq 2$.
Consider the function $f(x,a)=3x^2+2ax-1$. The maximum of this function is the same as for $|f|$. We can consider also a region: $$-2 \leq a \leq 2, \;\; -1 \leq x \leq 1$$ By the method which you do know that we find the origin as a interior critical point in which the function $f$ vanishes. Now, we consider the border which are $$x=-1,x=1,a=-2,a=2$$. A simple calculating shows us that at the point $(x,a)=(1,2)$ or $(-1,-2)$ the function reaches at $6$. and other points along side these borders are $$(2/3,-2),(-2/3,2)$$ which make the value $-7/3$. So the max would be $6$.