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Chapter 12: Q. 47 (page 986)

In Exercises 44鈥�49, find the maximum and minimum of the given function on the specified region. Also, give the points where the maximum and minimum occur.

f(x,y)=x2+yonsquarewithvertices(1,0),(0,1),(鈭�1,0),and(0,鈭�1)

Short Answer

Expert verified

The maximum value of the given function is 1at(0,1),(鈭�1,0)and(鈭�1,0)and the minimum value of the function is-1at0,-1.

Step by step solution

01

Step 1. Given Information.  

The given function is f(x,y)=x2+yand the given vertices are (1,0),(0,1),(鈭�1,0)and(0,鈭�1).

02

Step 2. Find the extreme of the function.

Let's find the gradient of the given function,

鈭�f(x,y)=2xi+j

To find the critical points put the gradient of the function equal to zero,

鈭�f(x,y)=2xi+j(0i+0j)=2xi+j2x=0x=0

Since the vertices are the boundary points.

Thus, the points where the minimum and maximum of the function appear are(1,0),(0,1),(鈭�1,0)and(0,鈭�1).

03

Step 3. Find the maximum and minimum of a function. 

Now, let's find the value of the function through the points.

So,

f(1,0)=1f(0,1)=1f(-1,0)=1f(0,-1)=-1

Thus, the maximum value of the function is 1which occurs at the points(0,1),(鈭�1,0)and(鈭�1,0) and the minimum value of the function is-1at0,-1.

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