When you are on top of a hill, you are at a local maximum although there may be other hills
higher than the one on which you are standing. Similarly, when you are at the bottom of a
valley, you are at a local minimum even though there may be other valleys deeper than the
one you are in. The word, “local” is applied to the situation because if you confine your
attention only to points close to your location, you are indeed at either the top or the
bottom.

Definition 7.6.1Let f : D

(f)

→ ℝ where here D

(f)

is only assumed to besome subset of F. Then x ∈ D

(f)

is a local minimum (maximum) if there exists δ > 0
such that whenever y ∈ B

(x,δ)

∩ D

(f)

, it follows f

(y)

≥

(≤ )

f

(x)

. The plural ofminimum is minima and the plural of maximum is maxima.

Derivatives can be used to locate local maxima and local minima. The following picture
suggests how to do this. This picture is of the graph of a function having a local maximum
and the tangent line to it.

PICT

Note how the tangent line is horizontal. If you were not at a local maximum or local
minimum, the function would be falling or climbing and the tangent line would not be
horizontal.

Theorem 7.6.2Suppose f : U → ℝ where U isan open subset of F andsuppose x ∈ U is a local maximum or minimum. Then f^{′}

(x)

= 0.

Proof: Suppose x is a local maximum and let δ > 0 is so small that B

(x,δ)

⊆ U. Then for

|h|

< δ, both x and x + h are contained in B

(x,δ)

⊆ U. Then letting h be real and
positive,

′
f (x) h+ o(h) = f (x+ h)− f (x) ≤ 0.

Then dividing by h it follows from Theorem 7.1.4 on Page 330,

( )
f′(x) = lim f′(x)+ o(h) ≤ 0
h→0 h

Next let

|h|

< δ and h is real and negative. Then

f′(x) h+ o(h) = f (x+ h)− f (x) ≤ 0.

Then dividing by h

f′(x) = lim f′(x)+ o(h) ≥ 0
h→0 h

Thus f^{′}

(x)

= 0. The case where x is a local minimum is handled
similarly. Alternatively, you could apply what was just shown to −f

Points at which the derivative of a function equals 0 are sometimes called critical points.
Included in the set of critical points are those points where f^{′} fails to exist.