This sounds very simple, and differentiability of f is not necessary.
What does it mean by definition that "f has a maximum value on an interval I at x=c"? That
for any `x in I,` `x != c`, `f(x)ltf(c)`
(this is for the strict maximum, and `f(x)lt=f(c)` for the not strict one).
Now for the function `-f(x).` Because `f(x)ltf(c)` for `x in I`, `x != c`, it is obvious that
`-f(x)gt-f(c)` for `x in I,` `x != c.`
To get this inequality, we multiplied the known inequality f(x)<f(c) by -1. It is legal, and both sides must be multiplied by -1 and the sign of the inequality must be changed to the opposite.
And this is the definition of a (strict) minimum for function -f(x), QED.