SOLUTION: Sketch the graph of the following functions by using the conditions of concavity f(x)=(x^2-16)^4

Algebra ->  Finance -> SOLUTION: Sketch the graph of the following functions by using the conditions of concavity f(x)=(x^2-16)^4       Log On


   

Question 1205395: Sketch the graph of the following functions by using the conditions of concavity f(x)=(x^2-16)^4
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29+=+%28x%5E2+-+16%29%5E4

find second derivative

f'%28x%29+=++8x+%28x%5E2+-+16%29%5E3
f''%28x%29+=+8+%28x%5E2+-+16%29%5E2+%287x%5E2+-+16%29+


So f ′′%28x%29=0 when
8+%28x%5E2+-+16%29%5E2+%287x%5E2+-+16%29+=0

critical points of the derivatives are:
x=4 => stationary
x=-4=> stationary
x=4%2Fsqrt%287%29=> stationary minimum
x=-4%2Fsqrt%287%29=> stationary maximum

so, f%28x%29+=+%28x%5E2+-+16%29%5E4 is
concave up on x%3C-4%2Fsqrt%287%29
concave down on -4%2Fsqrt%287%29%3Cx%3C4%2Fsqrt%287%29
concave up on x%3E4%2Fsqrt%287%29

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