Math Help

Examples involving Maxima and Minima of Functions on a Closed Domain

Example 1: Find the maximum and minimum of the function, f(x), given below in the

domain [ ˗ 1, 3 ].

f(x,y) = x² + 1

So df/dx = 2x = 0 and x = 0 is a critical point

At x = ˗ 1, f( ˗ 1) = 2 and at x = 3 f(3) = 10

So in the domain D, [ ˗ 1,3 ] the minimum of f(x) = 1 is at the critical point

and the maximum of f(x) = 10 is at the boundary point x = 3

Example 2: Find the maximum and minimum of the function, f(x), given below in the

domain [ ˗ 1, 2 ].

f(x) = 5 + | 3 ˗ 3 x |

If x > 1 , then f(x) = 5 + 3 x ˗ 3 = 2 + 3 x

If x < 1, then f(x) = 5 + 3 ˗ 3 x = 8 ˗ 3 x

Note: f(x) has a discontinuous derivative at x = 1

For x > 1, df/dx = 3 and for x < 1, df/dx = ˗ 3

Also note that f(1) = 5 f( ˗ 1) = 11, f(2) = 8

So the minimum of f(x) in the domain is 5 at x = 1

and the maximum of f(x) in the domain is 11 at x = ˗ 1

Example 3: Find the maximum and minimum of the function, f(x), given below in the

domain [ ˗ 2, 4 ].

f(x) = x³ ˗ 3 x² ˗ 9 x + 5

df/dx = 3 x² ˗ 6 x ˗ 9 = 0 for a max or a min

So 3 ( x² ˗ 2 x ˗ 3 ) = 0 or (x ˗ 3) ( x + 1) = 0

So the critical points are x = 3 and x = ˗ 1

Check all points on the boundary of the domain and within the domain.

Within the domain: f( ˗ 1) = 10 and f(3) = ˗ 22

On the boundary of the domain f( ˗ 2) = 3 and f(4) = ˗ 15

So the minimum of f(x) is ˗ 22 at x = 3 and the maximum of f(x) is 10 at x = ˗ 1