Find the curve y = f(x) in the xy-plane that has the given properties.
-$\frac { d ^ { 2 } y } { d x ^ { 2 } } = 36 x$ and the graph of $y$ passes through the point $( 0,11 )$ and has a horizontal tangent there.
A) $y = 18 x ^ { 3 } + 11$
B) $y = 18 x ^ { 2 } + 11$
C) $y = 6 x ^ { 3 } + 11$
D) $y = 6 x ^ { 3 } - 11$

Question

Quizplus · Accepted Answer

Integrating both sides of $\frac{d^2y}{dx^2}=36x$ twice, we get 
$$y=6x^3+C_1x+C_2 $$
where $C_1$ and $C_2$ are integration constants.

Using the fact that the graph passes through $(0,11)$, we have:
$$11=C_2$$

Also, the graph has a horizontal tangent at $(0,11)$. This means $\frac{dy}{dx}=0$ at $x=0$, which gives
$$0=18\cdot0^2+C_1$$
$$C_1=0$$

Therefore, the equation of the curve is 
$$y=6x^3+11$$

which matches with choice C.

Find the Curve Y = F(x) in the Xy-Plane That d2ydx2=36x\frac { d ^ { 2 } y } { d x ^ { 2 } } = 36 xdx2d2y​=36x

Find the Curve Y = F(x) in the Xy-Plane That $\frac { d ^ { 2 } y } { d x ^ { 2 } } = 36 x$