Relative Extrema Occur Only at Critical Numbers
(a.k.a. the Weierstrass extreme value theorem)

Recall the definition of a Critical Number:
Let f be defined at c.  If f'(c) = 0 or f' is undefined at c, then c is a critical number of f.

The statement "Relative Extrema Occur Only at Critical Numbers" means the same as:
"If f has a relative minimum or relative maximum at x=c then c is a critical number of f".

Proof:

Case 1: If f is not differentiable at c then c is a critical number, so the statement of this proof is true.

Case 2: If f is differentiable then f'(c) is positive, negative, or zero.

Suppose f'(c) is positive: if f'(c) > 0 then lim_x—>c (f(x)-f(c))/(x-c) > 0

so for all x ≠ c in some small interval (a,b) containing c, (f(x)-f(c))/(x-c) > 0

so when x is in (a,b) and x > c, it follows that f(x) > f(c)

and when x is in (a,b) and x < c, it follows that f(x) < f(c)

so c is not a relative maximum or a relative minimum, a contradiction, so f'(c) is not positive.

Suppose f'(c) is negative: if f'(c) < 0 then reasoning similar to case 2 leads to the conclusion that f'(c) is not negative.

Since f'(c) is neither positive nor negative, it must be zero, so c is a critical number of f, thus the statement of this proof is true.

How is this proof used?

Rolle's Theorem -- that if f(a)=f(b) then f'(c)=0 for some c in (a,b) -- depends on two key facts: that any function has a maximum on a closed interval (the Extreme Value Theorem), and that Relative Extrema Occur Only at Critical Numbers.

Related pages in this website

Definition of Critical Number -- a number, c, at which f' is undefined or f'(c)=0

Rolle's Theorem -- that if f(a)=f(b) then f'(c)=0 for some c in (a,b)

Relative Extrema Occur Only at Critical Numbers -- if c is an extremum then f'(c)=0 or f'(c) is undefined

The webmaster and author of this Math Help site is Graeme McRae.