The derivative of F is F'(x) = 4x^3 -16x = 4x * (x^2 -4)

For relative extrema, check the value of F where F’ = 0 on the interval [-3,3]. If allowed, also check F at the endpoints if your definition of RELATIVE extrema includes values at the endpoints.

F’ = 0 when 4x=0 ===> x=0

F’ = 0 when x^2 = 4 ===> x=+/-2

Relative extrema:

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F(0) = 0 – 0 + 24 = 24

F(2) = 16 – 32 + 24 = 8

F(-2) = 16 – 32 + 24 = 8

F(3) = 81 – 72 + 24 = 33 *** if the endpoints are allowed ***

F(-3) = 81 – 72 + 24 = 33 *** if the endpoints are allowed ***

Absolute extrema

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Absolute extrema are found by taking the relative extrema and the values at the end points.

F(2) and F(-2) are the absolute minimum value of this function with a value of 8.

F(-3) and F(3) are the absolute maximum value of this function on this interval with a value of 33.