What is the absolute minimum of $f$ if $f'(x) = (x - 7)(x + 1)(x + 5)$?

Question

Suppose a Calculus course has not taught anything about integrals yet. How can I tell which of the critical numbers $x=-5$, $x=7$ are lowest given the derivative in the question?

Answer 1

This is the first time I am uploading an answer from mobile phone, excited with fingers.

Shifting the polynomial $5$ units to the right we have $$g'(x)=x(x-4)(x-12)=x^3-16x^2+48x.$$ Assuming $g(0)=0$, we need to determine the sign of $g(12)$. It is enough to check the sign of $p(12)$ where $$p(x)=\frac{x^2}{4}-\frac{16}{3}x+\frac{48}{2}.$$ Now, $p(12)=-4<0$. Therefore, $x=12$ is the lower local minumum for $g(x)$ and $x=7$ is the lower local minumum for $f(x)$.