Given the function #f(x)=-(-5x+25)^(1/2_#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [3,5] and find the c?

Question

Christopher Agresta · Accepted Answer

See below.

The hypotheses af MVT are f is continuous on #[3,5]# f is differentiable on #(3,5)# If the both statements are true, then #f# satisfies the hypotheses of the Mean Value Theorem on the interval. To find the #c# mentioned in the conclusion of the theorem you need to find #f'(x)# and find #(f(5)-f(3))/(5-3)# and solve the equation: #f'(x) = (f(5)-f(3))/(5-3)#. Discard any solutions outside #(3,5)#.

Emilia Aguilar · Answer

undefined To determine if the function $ f(x) = -\sqrt{-5x + 25} $ satisfies the hypotheses of the Mean Value Theorem on the interval $[3, 5]$, we need to verify two conditions:

1. Continuity on the closed interval $[3, 5]$.
2. Differentiability on the open interval $(3, 5)$.

First, we check for continuity: $ f(x) $ is continuous everywhere inside the square root, and the square root function itself is continuous where it is defined. Hence, $ f(x) $ is continuous on the closed interval $[3, 5]$.

Next, we check for differentiability: $ f(x) $ is differentiable wherever the square root function is defined, except where its derivative would be undefined (such as when the expression under the square root becomes negative). In this case, the expression under the square root $ -5x + 25 $ is defined for all $ x $ in the interval $[3, 5]$, so $ f(x) $ is differentiable on the open interval $(3, 5)$.

Since $ f(x) $ satisfies the hypotheses of the Mean Value Theorem on the interval $[3, 5]$, we can find a $ c $ in the interval $(3, 5)$ such that $ f'(c) $ equals the average rate of change of $ f(x) $ over the interval $[3, 5]$.

To find $ c $, we first need to find $ f'(x) $ and then find the average rate of change of $ f(x) $ over the interval $[3, 5]$. Afterward, we can set $ f'(c) $ equal to this average rate of change and solve for $ c $.