Using the principle of the mean-value theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem for #f(x) = 5(1 + 2x)^(1/2)# in the interval [1,4]?

Question

Emma Aldridge · Accepted Answer

Solve the equation that is the conclusion to the Mean Value Theorem. Choose only the solutions in the interval #(1,4)# as the conclusion says.

Find #f'(x)#, then solve #f'(x) = (f(4)-f(1))/(4-1)#. The algebra is tedious. #f(4) = 15#, #f(1) = 5sqrt3# #f'(x) = 5/sqrt(1+2x)# So we need to solve: #5/sqrt(1+2x) = (15-5sqrt3)/3#. Which is equivalent to #1/sqrt(1+2x) = (3-sqrt3)/3# So, we need #sqrt(1+2x) = 3/(3-sqrt3) = (3+sqrt3)/2# #1+2x = (3+sqrt3)^2/4 = (6+3sqrt3)/2# So #x = (4+3sqrt3)/4 = 1 + (3sqrt3)/4#. Note that, since #3sqrt3 = sqrt 27 < 12#, this solution is in the interval #(1,4)#

Charles Anctil · Answer

undefined To find all numbers $ c $ that satisfy the conclusion of the Mean Value Theorem for $ f(x) = 5(1 + 2x)^{1/2} $ on the interval $[1,4]$, follow these steps:

1. Compute the derivative of $ f(x) $ with respect to $ x $.
$$ f'(x) = \frac{d}{dx}[5(1 + 2x)^{1/2}] $$

2. Evaluate $ f'(x) $ at each endpoint of the interval $[1,4]$.
$$ f'(1) \text{ and } f'(4) $$

3. Apply the Mean Value Theorem:
$$ f'(c) = \frac{f(b) - f(a)}{b - a} $$
where $ c $ is a number in the interval $[1,4]$ such that $ f'(c) $ equals the average rate of change of $ f(x) $ over the interval $[1,4]$.

4. Solve for $ c $ by setting $ f'(c) $ equal to the average rate of change of $ f(x) $ over the interval.
$$ f'(c) = \frac{f(4) - f(1)}{4 - 1} $$

5. Solve for $ c $.