How do you find the extreme points of the function f(x) = 0.25x4 + 3x3 – 18x2 + 10 and classify them?

To determine the extreme points of the function, we need
to determine the critical values first. The critical values are the roots of the 1st
derivative of the function.

For this reason, we'll
calculate the first derivative of the function:

f'(x) =
4*(1/4)*x^3 + 9x^2 - 36x

We'll cancel
f'(x):

f'(x) = 0

x^3 + 9x^2 -
36x = 0

We'll factorize by
x:

x*(x^2 + 9x - 36) = 0

We'll
cancel each factor:

x = 0

x^2
+ 9x - 36 = 0

We'll apply quadratic
formula:

x1 = [-9+sqrt(81 +
144)]/2

x1 = (-9+15)/2

x1 =
3

x2 = -12

The critical points
of the function are: x = 0 , x = 3 and x = -12.

To find
extreme points, we'll have to determine the y coordinates for the critical
values:

f(0) = 10

f(3) = 81/4
+ 81 - 162 + 10

f(3) = -71 +
81/4

f(3) = -203/4

f(-12) =
5184 + 5184 - 2592 + 10

f(-12) =
7786

The extreme values are: (0 , 10) ; (3 ,
-203/4) ; (-12 , 7786).