Question

Apply the mean value Theorem to f(x)=In(1+x) to show that
x1+x0. I am still unsure of this, I think I need to substitute any x>0 for each of the 3 equations and their derrivative x1+x 0 each equation will be less than the last?

Answer 1

\[f(x)=\ln(+x)\] is the original equation and I have to use the mean value theorem to show that\[\frac{ x }{1+x } < \ln(1+x) < x, \] for x > 0.

Answer 2

I'm on mobile and can't do any latex or drawing



So, to solve note simple statement of MVT: f'(c) = [ f(b) - f(a) ] / [b -a]

Using a = 0, b = x, calculate f'(c) in terms of x.

Then look separately at inequalities
(A) c< b, ie c< x and
(B) c> a,ie c> 0

Some faffing around and you will get there

Answer 3

Thank you so much, all sorted now. Big help. Thanks