Given the function #f(x)=(x^2-9)/(3x)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?

Question

Evelyn Agard · Accepted Answer

See below.

You determine whether it satisfies the hypotheses by determining whether #f(x) = (x^2-9)/(3x)# is continuous on the interval #[1,4]# and differentiable on the interval #(1,4)#. You find the #c# mentioned in the conclusion of the theorem by solving #f'(x) = (f(4)-f(1))/(4-1)# on the interval #(1,4)#. Answers #f# is continuous on its domain, which includes #[1,4]# #f'(x) = (x^2+9)/(3x^2)# which exists for all #x != 0# so it exists for all #x# in #(1,4)# Therefore this function satisfies the hypotheses of the Mean Value Theorem on this interval. To find #c# solve the equation #f'(x) = (f(4)-f(1))/(4-1)#. Discard any solutions outside #(1,4)#. I believe that you should get #c = 2# (because #-2# is outside the interval.).

Emma Aldridge · Answer

undefined To determine if the function $ f(x) = \frac{{x^2 - 9}}{{3x}} $ satisfies the hypotheses of the Mean Value Theorem on the interval $[1, 4]$ and find $ c $, you need to check two conditions:

1. **Continuity**: $ f(x) $ must be continuous on the closed interval $[1, 4]$.
2. **Differentiability**: $ f(x) $ must be differentiable on the open interval $(1, 4)$.

First, check the continuity of $ f(x) $ on $[1, 4]$ by verifying that there are no points of discontinuity within this interval.

Next, check the differentiability of $ f(x) $ on $(1, 4)$ by verifying that the derivative $ f'(x) $ exists and is continuous on $(1, 4)$. Compute $ f'(x) $ by differentiating $ f(x) $ with respect to $ x $.

Once you have confirmed continuity and differentiability, apply the Mean Value Theorem to find $ c $. The theorem states that if $ f(x) $ satisfies the hypotheses, there exists at least one $ c $ in the interval $(1, 4)$ such that $ f'(c) = \frac{{f(4) - f(1)}}{{4 - 1}} $.

Calculate $ f(4) $ and $ f(1) $, then find $ f'(c) $. Finally, solve for $ c $.