Find the function with the given derivative whose graph passes through the point P.
-$f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,12 )$
A) $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 12$
B) $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x + 12$
C) $f ( x ) = x ^ { 3 } + 6 x + 12$
D) $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x$

Question

Quizplus · Accepted Answer

To find the original function, we need to integrate the given derivative with respect to x. 
\begin{align*}
\frac{d}{dx}f(x) &= x^2 + 6 \
\int \frac{d}{dx}f(x) dx &= \int x^2 + 6 dx \
f(x) &= \frac{x^3}{3} + 6x + C
\end{align*}
where C is the constant of integration. To find the specific function that passes through the given point P(0,12), we can substitute these values into the equation and solve for C.
\begin{align*}
12 &= \frac{0^3}{3} + 6(0) + C \
C &= 12
\end{align*}
Thus, the final function is $f(x) = \frac{x^3}{3} + 6x + 12$. Choice B matches this function, so it is the correct answer.

Find the Function with the Given Derivative Whose Graph Passes f′(x)=x2+6,P(0,12)f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,12 )f′(x)=x2+6,P(0,12)

Find the Function with the Given Derivative Whose Graph Passes $f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,12 )$