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.

 

[M98Ranger]

It seems like you have correctly found the critical point by taking the partial derivatives and setting them equal to zero. However, it's possible that there is more than one critical point in this region, and the one you found (x=0, t=0) is not the maximum. This could be the case if there are other critical points that satisfy the given conditions of {-2 ≤ x ≤ 2 , 0 ≤ t ≤ 1}.

To find the maximum, you can use the second derivative test to determine if the critical point you found is a maximum or a minimum. If it is a maximum, then it should be the answer given in the back of the book. If it is a minimum, then there must be another critical point that is the maximum.

Another possibility is that there is a mistake in the given conditions or in the answer in the back of the book. It's always a good idea to double check the problem and the answer to make sure they are correct.

In summary, it seems like you have correctly found the critical point, but there may be other critical points or a mistake in the problem or answer. Keep exploring and checking your work to find the correct solution.