Q. 39 - Calculus

Question

For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

[a, b] = [0, 4]

Step-by-Step Solution

Answer

The hypothesis of Rolle's theorem is satisfied because from the graph of f appears to be continuous on $\left[0, 4\right]$ and differentiable on $\left(0, 4\right)$. The values of c that satisfy the conclusion of Rolle's theorem are $c\approx 0.5, c=2, c\approx 3.5$.

Step 1. Given information.

Consider the graph of f for the interval $\left[0, 4\right]$.

Step 2. Satisfy hypothesis of Rolle's theorem.

It can be observed that the given graph has no break, hole or gap in the interval $\left[0, 4\right]$. So, the graph of f appears to be continuous on $\left[0, 4\right]$.

It can be observed that the graph has no corner, no vertical line or no discontinuous point in the interval $\left(0, 4\right)$. So, the graph of f appears to be differentiable on $\left(0, 4\right)$.

Thus, the hypothesis of Rolle's theorem is satisfied. 

Step 3. Find values of c .

From the graph, $f\left(0\right)=0$ and $f\left(4\right)=0$.

$\Rightarrow f\left(0\right)=f\left(4\right)$

So, Rolle's theorem applies. There exists some $c\in \left(0, 4\right)$ such that $f\text{'}\left(c\right)=0$ or the graph has horizontal tangent line.

From the graph, such values of c where the graph has horizontal tangent line are $c\approx 0.5, c=2, c\approx 3.5$.