Chapter 3:

Applications of Derivatives

Examples of what can go wrong:

The hypotheses of the Extreme Value Theorem and the Mean Value Theorem both insist that f(x) be continuous on a closed interval.  Here are some examples to show why that condition is necessary.

How the extreme value theorem can "fail:"

1. Continuous, but interval is not closed: no maximum value:

f(x) = 1/(x-1) on (1 , 3]  The end point can be a vertical asymptote for the function:

2.  Continuous, but interval not closed: Does not achieve maximum:

f(x) = 1/x on (1 , 3].  End point would be maximum, but it isn't included:

3.  Not continuous but interval closed: No maximum value:

f(x) = 1/(x - 2)² on [1 , 3].  Contains asymptote in interior.

 4.  Not continuous but interval closed: Does not achieve maximum:

f(x) = { 2/((x-2)²+1) if x does not = 2, 1 if x = 2. on [1, 3]

How the max-min theorem can "fail:"

1.  Discontinuous at an end point:

f(x) = {1/x, if x is not 1, 0 if x is 1, on [1 , 3].

Slope of secant line is positive, slopes of tangents are all negative:

2.  Discontinuous in interior:

f(x) = [| x |] on [1 , 3]  (The "greatest integer" step function)

"Instantaneous change of position"

3.  Rolle's Theorem and the Mean Value Theorem also contains the additional proviso that the function be differentiable in the interior.  Here's an example of why that's necessary:

"Instantaneous change in velocity"

f(x) = 1 - | x - 2 | on [1 , 3]  .  Although the average velocity is zero, there is no place where the instantaneous velocity is zero....

Next Time, Section 3.3 (The First Derivative Test):