Question

A function that satisfies the hypothesis, and therefore the

conclusion, of the Mean Value Theorem.

Short answer

At point$\left(1,4\right)$the function satisfied the conditions of Rolle's theorem .

Step-by-step solution

Step 1. Given information .

A function that satisfies the hypothesis, of the Rolle's theorem .

Step 2. ln (x) is indeed continuous on [1,4] and differentiable on (1,4) therefore it satisfies the hypothesis of the mean value theorem. 

The Mean value theorem stated that the secant line connecting the points $\left(x_{1}f\left(x_1\right)\right),\left(x_{2}f\left(x_2\right)\right)$equals the slope of the tangent line at some c in the open interval $\left(x_1,x_2\right)$.

The slope of the secant line role="math" localid="1648627193201" $m=\frac{In\left(4\right)-In\left(1\right)}{4-1}=\frac{In\left(4\right)}{3}$.

$f\text{'}\left(x\right)=\frac{1}{x}$

Setting the derivative equal to the slope of secant and solving for x .

$$\begin{gathered} \frac{1}{x}=\frac{In\left(4\right)}{3} \\ x=\frac{3}{In\left(4\right)} \end{gathered}$$

Since this value$\frac{3}{In\left(4\right)}$of x indeed fall in open interval$\left(1,4\right)$so this satisfied the condition of Mean value theorem .