If is a function for which over an interval, then the graph of is convex upwards over this interval.
Likewise, if is a function for which over an interval, then the graph of is convex downwards over this interval.
A point of inflection on a curve is a point where the curve changes from convex downwards to convex upwards, or vice versa, so at a point of inflection, either , or is undefined.
First, use implicit differentiation to find the first derivative for the function as follows:
Now use implicit differentiation to find the second derivative of the function .
Next, since the graph of the function has an inflection point at , substitute 0 for , 3 for , and for in the equation and simplify.
Now substitute 3 for and for in the equation and simplify.
Next, to solve the system of equations , multiply the bottom equation by 2 to get the system , add the two equations to get , and divide both sides by 18 to get . Finally, substitute 8 for in the equation and solve for as follows:
Thus, the values of and that make the graph of the function have an inflection point at are 8 and 3, respectively.