Question

for the function ( f(x)=3x^{3}-2x^{2}+13x + 8 ), find ( f^{primeprime}(x) ). then find ( f^{primeprime}(0) ) and ( f^{primeprime}(9) ).
( f^{primeprime}(x)=)
select the correct choice below and fill in any answer boxes in your choice.
a. ( f^{primeprime}(0)=) (simplify your answer.)
b. ( f^{primeprime}(0) ) is undefined.
select the correct choice below and fill in any answer boxes in your choice.
a. ( f^{primeprime}(9)=) (simplify your answer.)
b. ( f^{primeprime}(9) ) is undefined.

Explanation:

Step1: Differentiate \(f(x)\) once

Use the power rule \((x^n)^\prime = nx^{n - 1}\).
For \(f(x)=3x^{3}-2x^{2}+13x + 8\), \(f^\prime(x)=3\times3x^{2}-2\times2x+13\times1+0=9x^{2}-4x + 13\).

Step2: Differentiate \(f^\prime(x)\) to get \(f^{\prime\prime}(x)\)

Differentiate \(f^\prime(x)=9x^{2}-4x + 13\) again.
\(f^{\prime\prime}(x)=9\times2x-4\times1+0 = 18x-4\).

Step3: Find \(f^{\prime\prime}(0)\)

Substitute \(x = 0\) into \(f^{\prime\prime}(x)\).
\(f^{\prime\prime}(0)=18\times0-4=-4\).

Step4: Find \(f^{\prime\prime}(9)\)

Substitute \(x = 9\) into \(f^{\prime\prime}(x)\).
\(f^{\prime\prime}(9)=18\times9-4=162 - 4=158\).

Answer:

\(f^{\prime\prime}(x)=18x - 4\); \(A. f^{\prime\prime}(0)=-4\); \(A. f^{\prime\prime}(9)=158\)