Use calculus to find the values of the constant $c$ for which the curve has local maximum and local minimum points. $g(x) = 4x^3 +cx^2 +10x$. Show that the graph always has one inflection point for every value of $c$, and determine the x-coordinate of the inflection point.
I think this has something to do with Fermat's Theorem ( if $f$ has a local max or min at $c$, and $f'(c)$ exists, then $f'(c) =0$) but I'm not sure how to set up the problem to solve it. Thank you!
$f’(x) = 12x^2 + 2cx+10=0$ must have two solutions for the existence of both a maximum and a minimum. So, the condition is $D \gt 0$ or $$4c^2 - 480 \gt 0$$
We also have $f’’(x) = 24x + 2c = 0 \implies x = \frac{-c}{12}$
This will have a solution for every $c$ and thus we get an inflection point in every case.