Question

for the function f(x)=5x³ - 3x² + 2x + 3, find f(x). then find f(0) and f(8).
f(x)=
select the correct choice below and fill in any answer boxes in your choice.
a. f(0)= (simplify your answer.)
b. f(0) is undefined.
select the correct choice below and fill in any answer boxes in your choice.
a. f(8)= (simplify your answer.)
b. f(8) is undefined.

Explanation:

Step1: Differentiate the function \( f(x) = 5x^{3}-3x^{2}+2x + 3 \)

Using the power rule \( \frac{d}{dx}(x^{n})=nx^{n - 1} \), we have:
\( f^{\prime}(x)=\frac{d}{dx}(5x^{3})-\frac{d}{dx}(3x^{2})+\frac{d}{dx}(2x)+\frac{d}{dx}(3) \)
\( f^{\prime}(x)=5\times3x^{2}-3\times2x + 2\times1+0 \)
\( f^{\prime}(x)=15x^{2}-6x + 2 \)

Step2: Find \( f^{\prime}(0) \)

Substitute \( x = 0 \) into \( f^{\prime}(x) \):
\( f^{\prime}(0)=15\times0^{2}-6\times0 + 2 \)
\( f^{\prime}(0)=2 \)

Step3: Find \( f^{\prime}(8) \)

Substitute \( x = 8 \) into \( f^{\prime}(x) \):
\( f^{\prime}(8)=15\times8^{2}-6\times8 + 2 \)
\( f^{\prime}(8)=15\times64-48 + 2 \)
\( f^{\prime}(8)=960-48 + 2 \)
\( f^{\prime}(8)=914 \)

Answer:

\( f^{\prime}(x)=15x^{2}-6x + 2 \); \( A. f^{\prime}(0)=2 \); \( A. f^{\prime}(8)=914 \)