How to Find $a$ and $b$ Such That the Function $f(x) = ax^3+bx^2$ has an Inflection Point at $ (2,64)$ I am trying to figure out how to find $a$ and $b$ of $f(x)$, such that the function $f(x) = ax^3+bx^2$. Also, $f(x)$ has an inflection point at $ (2,64)$.
Now I know that $f(2)$ = 64, however, I am unsure how to find $a$ and $b$ if $64 = 8a + 4b$. For example, $a$ could equal 4, and $b$ = 8, and the function would still equal $f(x)$. Although, this is apparently incorrect - does anyone know why, and how I would figure out what values $a$ and $b$ are?

An inflection point must occur at a zero of the second derivative. So just compute $f''$ and set $f''(2) = 0$. Then, combined with your condition $f(2) = 64$, you'll have two equations in the two variables $a$ and $b$, so you'll be able to solve for them. Then check that $x = 2$ actually is an inflection point, i.e. $f''$ changes sign at $x = 2$.
