140 CHAPTER 7. THE DERIVATIVE

Definition 7.6.1 Let f : D( f )→ R where here D( f ) is only assumed to be somesubset of F. Then x ∈ D( f ) is a local minimum (maximum) if there exists δ > 0 such thatwhenever y∈B(x,δ )∩D( f ), it follows f (y)≥ (≤) f (x) . The plural of minimum is minimaand the plural of maximum is maxima.

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

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

Theorem 7.6.2 Suppose f : U → R where U is an open subset of F and supposex ∈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. Thenfor |h| < δ , both x and x+ h are contained in B(x,δ ) ⊆ U . Then letting h be real andpositive,

f ′ (x)h+o(h) = f (x+h)− f (x)≤ 0.

Then dividing by h it follows from Theorem 7.1.5 on Page 128,

f ′ (x) = limh→0

(f ′ (x)+

o(h)h

)= lim

h→0

(1h( f (x+h)− f (x))

)≤ 0

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) = limh→0

f ′ (x)+o(h)

h= lim

h→0

(1h( f (x+h)− f (x))

)≥ 0

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 (x). 1

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.

7.7 Exercises1. If f ′ (x) = 0, is it necessary that x is either a local minimum or local maximum?

Hint: Consider f (x) = x3.

1Actually, the case where the function is defined on an open subset of C and yet has real values is not toointeresting. However, this is information which depends on the theory of functions of a complex variable whichis not considered yet.