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 limx-->c
(f(x)-f(c))/(x-c) > 0
so for all x ¹ 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
