Question

The point (3, 1/4) is on the graph of
y=f(x), and the slope at each pointy (x,y) on the graph is given by dy/dx=y^2(6-2x). Find d^2y/dx^2 and evaluate it at the point (3, 1/4).

Answer 1

since:
\[\frac{dy}{dx}=y^2(6-2x) \rightarrow \frac{dy}{dx}=6y^2-2y^2x\]
\[\frac{d^2y}{dx^2}=12y \frac{dy}{dx}-4yx \frac{dy}{dx}-2y^2\]
\[\rightarrow \frac{d^2y}{dx^2}=(12y-4yx)\frac{dy}{dx}-2y^2\]
now substitute dy/dx in...
\[\rightarrow \frac{d^2y}{dx^2}=(12y-4yx)(6y^2-2y^2x)-2y^2\]
now plug in 3 for x and 1/4 for y.... let me know what you get

Answer 2

you should get -1/8