Q. 44 - Calculus

Question

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem. 

$f\left(x\right)=\frac{x^2-4x}{x^2-4x+3}, \left[a, b\right]=\left[0, 4\right]$

Step-by-Step Solution

Answer

The function $f\left(x\right)=\frac{x^2-4x}{x^2-4x+3}$ does not satisfies the hypotheses of Rolle's theorem.

Step 1. Given information.

Consider the given function $f\left(x\right)=\frac{x^2-4x}{x^2-4x+3}, \left[a, b\right]=\left[0, 4\right]$.

Step 2. Satisfy hypotheses of Rolle's theorem.

Simplify the function.

$f\left(x\right)=\frac{x\left(x-4\right)}{\left(x-3\right)\left(x-1\right)}$

Find the values where function is not defined.

$\begin{array}{rcl}\left(x-3\right)\left(x-1\right)& =& 0\\ x& =& 3, 1\end{array}$

So, the function is not continuous at $x=3,1$ in the interval $\left[0, 4\right]$.

Thus, the given function des not satisfy the hypotheses of Rolle's theorem.