Find the function with the given derivative whose graph passes through the point P.
-$$g ^ { \prime } ( x ) = \frac { 4 } { x ^ { 2 } } + 6 x , P ( - 2,5 )$$
A) $g ( x ) = - 4 x ^ { - 1 } + 3 x ^ { 2 }$
B) $g ( x ) = - 4 x ^ { - 1 } + 3 x ^ { 2 } - 9$
C) $g ( x ) = 4 x ^ { 2 } + 3 x ^ { 2 } - 9$
D) $g ( x ) = 4 x ^ { - 2 } + 6 x - 9$

Question

Quizplus · Accepted Answer

Integrating the given derivative we get:
$$g(x) = -4x^{-1} + \frac{3}{2}x^2 + C$$ where $C$ is a constant of integration.
Since the graph passes through point $P(-2,5)$, we can use it to determine the value of $C$:
$$5 = -4(-2)^{-1} + \frac{3}{2}(-2)^2 + C$$
$$5 = -2 + 6 + C$$
$$C = -9$$
Substituting this value of $C$ in the equation for $g(x)$ we get:
$$g(x) = -4x^{-1} + \frac{3}{2}x^2 - 9$$
Therefore, the answer is choice B.

Find the Function with the Given Derivative Whose Graph Passes g′(x)=4x2+6x,P(−2,5)g ^ { \prime } ( x ) = \frac { 4 } { x ^ { 2 } } + 6 x , P ( - 2,5 )g′(x)=x24​+6x,P(−2,5)

Find the Function with the Given Derivative Whose Graph Passes $g ^ { \prime } ( x ) = \frac { 4 } { x ^ { 2 } } + 6 x , P ( - 2,5 )$