let the function f be differentiable on an interval I containing c. If f has a maximum value at x=c, show that -f has a minimum value at x=c. Please explain how to solve this question.

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Borys Shumyatskiy eNotes educator| Certified Educator


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.