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

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crmhaske | College Teacher | (Level 3) Associate Educator

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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

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