Proof involving the mean value theorem and derivatives

[imurme8]

Homework Statement

For $\mu\geq 0, s\geq 1,$ prove that $(1+s)^{\mu}\geq 1 + s^{\mu}$

Homework Equations

The Attempt at a Solution

I have written a proof involving the mean value theorem and derivatives, but there must be a simpler way! I think this should be done purely algebraically. Instructor insists that $\mu$ is an arbitrary non-negative real number, not just a rational or an integer. So to define it we need log, etc. But I believe there is a solution that does not go into this...

 

[imurme8]

Forgot to say, no binomial theorem allowed

 

[imurme8]

Whoops, meant $\mu \geq 1$ and $s\geq 0$, not the other way around. However, they are both reals.

 

[LCKurtz]

Homework Statement

For $\mu\geq 0, s\geq 1,$ prove that $(1+s)^{\mu}\geq 1 + s^{\mu}$

What does this question have to do with the subject "ddd" of the thread?

 

[imurme8]

Whoops, meant $\mu \geq 1$ and $s\geq 0$, not the other way around. However, they are both reals.

 

[imurme8]

What does this question have to do with the subject "ddd" of the thread?

I apologize, I forgot to enter a thread name that makes sense. If we can delete this one and repost it, I'd be happy to.

 

[berkeman]

I apologize, I forgot to enter a thread name that makes sense. If we can delete this one and repost it, I'd be happy to.

(I changed the thread title for you.)