A function f(x) and interval [a,b] are given. Check if the Mean Value Theorem can be applied to f on [a,b]. If so, find all values c in [a,b] guaranteed by the Mean Value Theorem f(x)=5lnx−1 on [1,10]

Question

Note, if the Mean Value Theorem does not apply, enter DNE for the  value.

f(x)=5lnx−1 on [1,10]

c=

 

 

(Separate multiple answers by commas.)

Please help! I don't know how to solve!

Answer 1

f(x) needs to be continuous on [1,10] which it is

It needs to be differentiable on (1,10)

f'(x) = 5/x , this exists on (1,10) so differentiable

So for some c in (1,10), f'(c) = f(10) - f(1) / (10-1)

f(10) - f(1) = 5 ln 10

So we need f'(c) = 5 ln 10 / 9

5/x = 5 ln 10 / 9

x = 9 / ln 10 = 3.91