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]

Short 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 ofc that satisfy the conclusion of Rolle's theorem are$c\approx 0.5,c=2,c\approx 3.5$.

Step-by-step solution

Step 1. Given information.

Consider the graph off 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 $c\approx 0.5,c=2,c\approx 3.5$.