Graph, figure and maximum and minimum

[Niles]

Homework Statement

Please take a look at the attached figure.

It is the function f(x,y) = x^2*y in the interval -1 =< x =< 1 and x =< y =< x+2.

From the figure, is it correct that the minimum is in f(-1,-1) = -1 and the maximum is in f(1,3) = 3?

 

[Niles]

The level curve is plottet as well

 

[Niles]

Do I have to give 2 or 3 coordinates?

 

[HallsofIvy]

Since f is a function of 2 variables, you specify the point at which f has a maximum or minimum by giving the x and y values of the point. The third variable shown in the graph is the value of f. In order to determine the maximum and minimum values, you can take the gradient of f inside the region shown and determine if there are any (x,y) points where it is 0. Then look on the boundaries. Since the problem says, " in the interval -1 =< x =< 1 and x =< y =< x+2" (that's not actually an interval, by the way. It should say "region".) The boundaries are x= -1, with y between -1 and 1 (replace x in the formula by -1 to get a function of y only), y= x with x between -1 and 1 (replace y in the formula by x to get a function of x only), y= x+2 with x between -1 and 1 (replace y in the formula by x+ 2 to get a function of x only), and x= 1 with y between 1 and 3 (replace x in the formula by 1 to get a function of x only). Again, take the derivative to find if there are any places where it is 0. Finally, the boundaries of those- the corners of the region, (-1, -1), (-1, 1), (1, 1), and (1, 3). Check the value of f at all of those points to see which gives the maximum value and which gives the minimum value.