Find the extreme point(s) of the function f(x)=.25x^4 +3x^3-18x^2 +10 and classify them

To locate the extreme points of a function you need to
take the first derivative and set it equal to zero.  The first deravtive tells you where
the function changes from increasing to decreasing or vice versa.  To determine whether
the point is a maximum or a minimum you take the second dervative and sub in your
points.  If the second derivative is negative the extreme is a maximum, if the second
derivative is positive the exterme is a minimum.

First
derivative:
f'(x)=0 = x^3+9x^2-36x =
x(x^2+9x-36)=x(x+12)(x-3)

Through factoring the first
derivative we see that the extremes are located at x=0, x=-12, and
x=3

Second derivative:
f''(x) =
3x^2+18x-36

f''(0) = -36 ... therefore
maximum
f''(-12) = 180 ... therefore minimum
f(3) = 45 ... therefore
minimum