For the function below, determine each of the following. f(x) = 3x4 + 96x2 24 (a

Question

For the function below, determine each of the following. f(x) = 3x4 + 96x2 24 (a) Find all critical values of f(x). (Enter your answers as a comma-separated list.) x = (b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing. Enter your answers using interval notation. Increasing: Decreasing: (c) Find the x-coordinates of all relative extrema on the graph of f(x). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Relative Maxima: Relative Minima:

Explanation / Answer

f(x) = 3x4 + 96x2 24

for critical points f '(x)=0

=> f '(x) = 3*4x3 + 96*2x 0=0

-12(x3-16x)=0

-12(x3-16x)=0

-12x(x2-16)=0

-12x(x-4)(x+4)=0

-12(x+4)x(x-4)=0

x =-4,x=0,x=4

critical points are x =-4,x=0,x=4

b)function is increasing if f '(x)>0

=>-12(x+4)x(x-4)>0

=>x=(-infinity,-4)U(0,4)

function is decreasing if f '(x)<0

=>-12(x+4)x(x-4)<0

=>x=(-4,0)U(4,infinity)

c)f '(x) changes from negative to positive at x =0

so relative minima at x =0

f '(x) changes from psitive to negative at x =-4,4

so relative minima at x =-4,4

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