Question

for the function $f(x)=2x^3 - 3x^2 + 5x + 3$, find $f(x)$. then find $f(0)$ and $f(3)$. to find $f(x)$, first find $f(x)$. $f(x)=\square$

Explanation:

Step1: Apply Power Rule for Differentiation

The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n - 1} \). We apply this to each term of \( f(x)=2x^{3}-3x^{2}+5x + 3 \).

For the term \( 2x^{3} \): Using the power rule, the derivative is \( 2\times3x^{3 - 1}=6x^{2} \).

For the term \( - 3x^{2} \): The derivative is \( -3\times2x^{2 - 1}=-6x \).

For the term \( 5x \): The derivative is \( 5\times1x^{1 - 1}=5 \) (since \( x^{0}=1 \)).

For the constant term \( 3 \): The derivative of a constant is \( 0 \).

Step2: Combine the Derivatives of Each Term

Adding up the derivatives of each term, we get:
\( f'(x)=6x^{2}-6x + 5+0=6x^{2}-6x + 5 \)

Answer:

\( 6x^{2}-6x + 5 \)