Question

Find the absolute maximum and minimum values of the following functions on the given curves functions:

a. f(x,y)= x+y
b. g(x,y)= xy
c. h(x,y)= 2x^2+y^2

Answer 1

Solution has explained below:

Explanation:

(a) f(x,y) = x+y

To find out the maximum and minimum values, we need to find first and second derivatives, we have

fx= 1, fx₁=0 and

fy= 1 and fyy=0

For stationary points fx=fy=0, which gives,

1=1=0, so that there is just one stationary point, (x,y)=(0,0)

If fx₁ < 0 and fyy < 0, function is maximum

If fx₁ > 0 and fyy > 0, function is minimum.

(b) g(x,y) = xy

Sol: To find out the maximum and minimum values, we need to find first and second derivatives, we have

fx= 1 and fy = 1

fx₁ = 0 and fyy =0

for stationary points fx = fy =0, which gives 1=1=0, so that there is just one stationary points, (x,y) =(0,0)

If fx₁ < 0 and fyy < 0, function is maximum.

If fx₁ > 0 and fyy> 0, function is minimum.

c) h(x,y) = 2x2 + y2

sol: To find out the maximum and minimum values, we need to find first and second derivatives, we have

fx = 4x and fy = 2y

fx₁= 4 and fyy = 2

Now, taking fx = 0 and fy = 0

Gives x=0 and y=0,

Stationary points are (x,y) = (0,0)

If fx₁ < 0 and fyy < 0, solution is maximum,

If fx₁ > 0 and fyy > 0, solution is minimum,

Here, fx₁ = 4 > 0 and fyy = 2 > 0.