5 Steps to a 5 AP Calculus BC 2019

Graphs of Functions and Derivatives 107

TIP • The formula for finding the area of an equilateral triangle isarea=s

2 √ 3

4

wheresis the length of a side. You might need this to find the volume of a solid whose

cross sections are equilateral triangles.

Extreme Value Theorem

Iff is a continuous function on a closed interval [a,b], thenfhas both a maximum and

a minimum value on the interval.

Example 1

Iff(x)=x^3 + 3 x^2 −1, find the maximum and minimum values off on [−2, 2]. Sincef(x)

is a polynomial, it is a continuous function everywhere. Entery 1 =x^3 + 3 x^2 −1. The graph

ofy1 indicates thatfhas a minimum of−1atx=0 and a maximum value of 19 atx=2.

(See Figure 7.1-7.)

[–3,3] by [–4,20]

Figure 7.1-7

Example 2

If f(x)=

1

x^2

, find any maximum and minimum values of f on [0, 3]. Since f(x)isa

rational function, it is continuous everywhere except at values where the denominator is

0. In this case, atx=0, f(x) is undefined. Since f(x) is not continuous on [0, 3], the
   Extreme Value Theorem may not be applicable. Entery 1 =

1

x^2

. The graph ofy1 shows
that asx → 0 +, f(x) increases without bound (i.e., f(x) goes to infinity). Thus, f has
no maximum value. The minimum value occurs at the endpointx=3 and the minimum
value is

1

9

. (See Figure 7.1-8.)

[–1,4] by [–1,6]

Figure 7.1-8