Question

For the polynomial function ƒ(x) = 9 − x², find all local and global extrema.

Answer 1

The polynomial function is f( x ) = 9 - x².

Apply first derivative with respect to x.

f '( x ) = - 2x

Apply second derivative with respect to x.

f ''( x ) = - 2.

To find the critical values, or does not exist.

f '( x ) = - 2x = 0

⇒ x = 0.

Relative extrema :

Find Extrema :

To find out extrema, use theorem.

If f " (x) > 0 (positive) ------> minimum point.

If f " (x) < 0 (negative) ------> maximum point.

f ''( x ) = - 2 < 0(negative) ------> maximum point.

To find the f(x) to each x for max and min plugging those values in the original function.

If x = 0 then,

f(0) = 9 - (0)² = 9 - 0 =  9.

The relative maximum is f(0) = 9.

Answer 2

Minimum and Maximum Values of Quadratic Functions :

Consider the function  with vertex .

1. If a > 0, f(x) has a minimum at . The minimum value is .

2. If a < 0, f(x) has a maximum at . The maximum value is .

The polynomial function is .

Compare the polynomial function with general form of quadratic function .

a = - 1, b = 0 and c = 9.

Since a = - 1 < 0, f(x) has a maximum at  .

The maximum value is and this is called global maximum.

(Note: Quadratics open upward or downward, therefore they will never have any local maximum or minimums)

The leading coefficient (a = - 1) is negative hence f(x) has a maximum at (0 , 9) and f(x) is increasing on (- infinity, 9) and decreasing on (9, + infinity). See graph below of f(x) below.