Calculus- Early Transcendentals, 2021a

168 Applications of Derivatives
.
A
•
A
•
A
•
•
B
•
B
.
Figure 5.5: Some local maximum points (A) and minimum points (B).
If (x, f (x)) is a point where f (x) reaches a local maximum or minimum, and if the derivative of f
exists at x, then the graph has a tangent line and the tangent line must be horizontal. This is important
enough to state as a theorem.
The proof is simple enough and we include it here, but you may accept Fermat’s Theorem based on its
strong intuitive appeal and come back to its proof at a later time.
Theorem 5.8: Fermat’s Theorem
If f (x) has a local extremum at x = a and f is differentiable at a, then f ′ (a)=0.
Proof. We shall give the proof for the case where f (x) has a local maximum at x = a. The proof for the
local minimum case is similar.
Since f (x) has a local maximum at x = a, there is an open interval (c,d) with c < a < d and f (x) ≤
f (a) for every x in (c,d).So, f (x) − f (a) ≤ 0 for all such x. Let us now look at the sign of the difference
f (x) − f (a)
quotient . We consider two cases according as x > a or x < a.
x − a
If x > a,thenx − a > 0andso,
f (x) − f (a)
x − a
≤ 0. Taking limit as x approach a from the right, we get
f (x) − f (a)
lim
≤ 0.
x→a + x − a
f (x) − f (a)
On the other hand, if x < a,thenx − a < 0andso, ≥ 0. Taking limit as x approach a from the
x − a
left, we get
f (x) − f (a)
lim
≥ 0.
x→a − x − a
Since f is differentiable at a, f ′ f (x) − f (a) f (x) − f (a)
(a) = lim
= lim
. Therefore, we have both
x→a + x − a x→a − x − a
f ′ (a) ≤ 0and f ′ (a) ≥ 0. So, f ′ (a)=0.
♣
Thus, the only points at which a function can have a local maximum or minimum are points at which
the derivative is zero, as in the left hand graph in Figure 5.5, or the derivative is undefined, as in the right
hand graph. Any value of x in the domain of f for which f ′ (x) is zero or undefined is called a critical