Chapter 4, Problem 4RCC

(a)

To determine

To state: The Rolle ’s Theorem.

Explanation of Solution

The Rolle’s theorem states that, “if function f satisfies the following three hypotheses,

(1) Function f is continuous on $\left[a,b\right]$.

(2) Function f is differentiable on $\left(a,b\right)$

(3) $f\left(a\right)=f\left(b\right)$

Then, there is a number c in $\left(a,b\right)$ such that $f^{\prime }\left(c\right)=0$”.

(b)

To determine

To state: The mean value theorem and interpret its geometrical meaning.

Explanation of Solution

The mean value theorem states that, “if a function f satisfies the following hypotheses,

(1)Function f is continuous on $\left[a,b\right]$.

(2) Function f is differentiable on $\left(a,b\right)$

Then, there is a number c in $\left(a,b\right)$ such that $f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$”.

Geometrically mean value theorem can be explained as follow.

The slope of the secant line joining the points $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$ to the curve is, $\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

Therefore, the equation $\frac{f\left(b\right)-f\left(a\right)}{b-a}=f^{\prime }\left(c\right)$ represents the tangent to the function $y=f\left(x\right)$.

Diagrammatic representation of the function is shown below in Figure 1.

From Figure 1, it is noticed that secant line becomes tangent with slope $\frac{f\left(b\right)-f\left(a\right)}{b-a}$.