19 multiple choice 1 point find f such that the given conditions are satisfied. f(x) = x² - 7x + 11, f(0) = 6 f(x) = (1/3)x³ - (7/2)x² + 11x + 1 f(x) = (1/3)x³ - 8x² + 11x + 1 f(x) = 3x³ - 8x² + 11x + 6 f(x) = (1/3)x³ - (7/2)x² + 11x + 6

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Accepted Answer

# Explanation:
## Step1: Integrate $ f'(x) $
To find $ f(x) $ from $ f'(x) $, we integrate $ f'(x) = x^2 - 7x + 11 $ with respect to $ x $. Using the power rule for integration, $ \int x^n dx=\frac{x^{n + 1}}{n+1}+C $ ($ n
eq - 1 $):
$$
\begin{align*}
f(x)&=\int(x^{2}-7x + 11)dx\
&=\int x^{2}dx-7\int xdx+11\int dx\
&=\frac{x^{3}}{3}-7\times\frac{x^{2}}{2}+11x + C\
&=\frac{1}{3}x^{3}-\frac{7}{2}x^{2}+11x + C
\end{align*}
$$
## Step2: Find the constant $ C $
We know that $ f(0) = 6 $. Substitute $ x = 0 $ into the expression for $ f(x) $:
$$
\begin{align*}
f(0)&=\frac{1}{3}(0)^{3}-\frac{7}{2}(0)^{2}+11(0)+C\
6&=0 - 0+0 + C
\end{align*}
$$
So, $ C = 6 $.
## Step3: Write the final function $ f(x) $
Substitute $ C = 6 $ back into the expression for $ f(x) $:
$$
f(x)=\frac{1}{3}x^{3}-\frac{7}{2}x^{2}+11x + 6
$$

# Answer:
$ f(x)=\frac{1}{3}x^{3}-\frac{7}{2}x^{2}+11x + 6 $ (the fourth option)

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QUESTION IMAGE

Question

Explanation:

Step1: Integrate \( f'(x) \)

Step2: Find the constant \( C \)

Step3: Write the final function \( f(x) \)

Answer: