Maximum of a function u(x,t)

[forget_f1]

I have u(x,t)=-2xt-x^2 find maximum in region {-2 ≤ x ≤ 2 , 0 ≤ t ≤ 1}

I believe to find the critical point first I have to take the partial derivative with respect to x and t and equate to zero.
Thus
Ux=-2t-2x = 0
Ut=-2x = 0

Thus the only critcal point I find is x=0, t=0.
But the maximum (answer at back of book) is x=-1, t=1 => u(-1,1)=1

Where did I go wrong?

[Tide]

That's because (0, 0) is a saddle point (check the second derivatives!). You need to examine the absolute maximum of the function in the region.

[forget_f1]

Ok, got it. I forgot that then I need to evaluate along the boundaries of the region.