Consider the function $f(x) = px^3 - q^2x$ where $p$ and $q$ are positive constants. Find all $x$-values where $f$ has local minima (if any).
I graphed this function and there was a local minima at $x = 0,577$ and lots of other values when $p$ and $q$ were changed. How do I find all of them ?
For a general case with $p,q$ being positive constants, you should differentiate the function and find all its critical points.
$$f(x)=px^{3}-q^{2}x \Rightarrow f^{\prime}(x)=3px^{2}-q^{2}$$ Then to find critical points, $$f^{\prime}(x)=0, 3px^{2}-q^{2}=0$$ By rearranging the equation, you get $$x^{2}=\frac{q^{2}}{3p}$$ Since $p$ is positive, $p \neq 0$. Square rooting both sides will get you the values of the critical points of $f$. $x= \pm \frac{q}{\sqrt{3p}}$.
To conclude, the local extremas are $(\frac{q}{\sqrt{3p}},f(\frac{q}{\sqrt{3p}}))$ and $(-\frac{q}{\sqrt{3p}},f(-\frac{q}{\sqrt{3p}}))$. You can do the algebraic manipulations yourself. To test which is the minima and maxima, you can apply the second derivative test to both of these points to determine it.