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

1 Answer
Jim H
May 24, 2017

Please see below.

Explanation:

The Mean Value Theorem has two Hypotheses.
A hypothesis is satisfied if it is true.

H1: The function must be continuous on #[1,4]#.

So, ask your self where #f# is NOT continuous. Are there any numbers in #[1,4]# where #f# is not continuous?

H2: The function must be differentiable on #(1,4)#.

Find #f'# and determine whether there are any numbers in #(1,4)# where #f'# is not defined.

To (try to) find the #c# mentioned in the conclusion of the Mean Value Theorem,

set #f'(x) = (f(4)-f(1))/(4-1)# and solve the resulting equation.

Since the conclusion asserts the existence of a #c# in the interval #(1,4)# any solutions to the equation that are in the interval are values for #c#.

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