What is the Maximum Value of u(x,t) in the Given Region?

In summary, when finding the maximum of the function u(x,t)=-2xt-x^2 in the region {-2 ≤ x ≤ 2, 0 ≤ t ≤ 1}, it is important to take the partial derivatives with respect to both x and t and equate them to zero. However, this only gives the critical point (0,0), which is a saddle point. To find the absolute maximum, it is necessary to also evaluate the function along the boundaries of the given region. In this case, the absolute maximum is found to be u(-1,1)=1.
  • #1
forget_f1
11
0
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?
 
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  • #2
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.
 
  • #3
Ok, got it. I forgot that then I need to evaluate along the boundaries of the region.
 

FAQ: What is the Maximum Value of u(x,t) in the Given Region?

What is the maximum of a function?

The maximum of a function is the highest value that the function can reach within a given domain.

How is the maximum of a function calculated?

The maximum of a function can be calculated by finding the critical points of the function, where the derivative is equal to zero, and evaluating the function at those points to determine the maximum value.

Can a function have multiple maximum values?

No, a function can only have one maximum value. However, it is possible for a function to have multiple local maximum values within different intervals of its domain.

How is the maximum of a function affected by its parameters?

The maximum of a function can be affected by its parameters, as changing the parameters can shift the position of the maximum value or change its value altogether.

Why is finding the maximum of a function important in science?

Finding the maximum of a function is important in science because it allows us to determine the optimal value of a variable in a given situation. This can be useful in many fields, such as optimization problems in engineering or determining the peak concentration of a chemical in a reaction.

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