Question

Find the absolute maximum and minimum values of f on the given interval. f(x)=4x^{3}-6x^{2}-24x+9. [-2,3]

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asked 2021-05-09
Find the absolute maximum and minimum values of f on the given interval.
\(f(x)=4x^{3}-6x^{2}-24x+9. [-2,3]\)

Answers (1)

2021-05-10
Step 1
We will find the first derivative using the power rule
\(f(x)=4x^{3}-6x^{2}-24x+9\)
\(f'(x)=4(3x^{2})-6(2x)-24\)
\(f'(x)=12x^{2}-12x-24\)
Step 2
We find the critical values by solving f'(x)=0
\(12x^{2}-12x-24=0\)
\(12(x^{2}-x-2)=0\)
12(x-2)(x+1)=0
x=2,-1
Then we find the values of the function at the critical points and at the endpoints.
\(f(x)=4x^{3}-6x^{2}-24x+9\)
\(f(-2)=4(-2)^{3}-6(-2)^{2}-24(-2)+9=1\)
\(f(-1)=4(-1)^{3}-6(-1)^{2}-24(-1)+9=23\) (max)
\(f(2)=4(2)^{3}-6(2)^{2}-24(2)+9=-31\) (min)
\(f(3)=4(3)^{3}-6(3)^{2}-24(3)+9=-9\)
Answer:
Absolute minimum value= -31
Absolute maximum value=23
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