a confusing prob. on maxima and minima

[heman]

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?

[sal]

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?
You looked for a global extremum. There's just one, as you found. But you overlooked the constraint -- you're only looking at a little piece of the domain. For a simpler example that shows the same issue, look at

y = 2x \ \ \ \lbrace 0 \leq x \leq 1 \rbrace

Its derivative wrt x is 2 ... never zero. But it certainly has a maximum on the region [0,1], at 1, where it takes the value y=2.

When the problem is constrained you need to look for a maximum or minimum in the interior of the domain, as you did, but then you also need to work your way around the boundary looking for maxima and minima there, as well.