Q. 75 - Calculus

Question

Explain in practical terms what the Extreme Value Theorem says about each continuous function defined in Exercises $75-77$. Then explain in practical terms what the Intermediate Value Theorem says in each situation.

Alina hasn’t cut her hair for six years. Six years ago her hair was just $2$ inches long. Now her hair is $42$ inches long. Let $H\left(t\right)$ be the function that describes the length, in inches, of Alina’s hair $t$ years after she stopped cutting it. 

Step-by-Step Solution

Answer

The Extreme Value Theorem says Alina must have had a maximum length and a minimum length in six years.

The Intermediate Value Theorem tells us that Alina's hair has been every length between $2$ and $42$ inches at some time during the last six years. 

Step 1. Given Information

Length of Alina's hair six years ago $=2$ inches

Length of Alina's hair  now $=42$ inches

Alina hasn’t cut her hair for six years. 

We need to find what the Extreme Value Theorem and Intermediate Value Theorem says in this situation.

Step 2. Extreme Value Theorem

The length function $H\left(t\right)$ should be continuous because a person’s hair length changes continuously over time and cannot jump from one value to another.

The Extreme Value Theorem tells us that there is some time in six years at which the Alina's hair length was longest and some time in six years at which that hair length was shortest. In other words, at some time in the six years, Alina must have had a maximum length and a minimum length. 

Step 3. Intermediate Value Theorem

The Intermediate Value Theorem tells us that for every hair length K between $2$ inches and $42$ inches, there is some time $c$ in six years for which $H\left(c\right)=K$.