Solving Simple Inequality: Tips/Suggestions Needed

[autobot.d]

Is there a way to do this without differentiation?

\left(a+b\right)^{p} \leq a^{p}+b^{p}


0<p<1 and a,b\geq 0

pulling the a out of the the first part and dividing by it to get

\left(1+\frac{b}{a}\right)^{p}\leq 1+\frac{b}{a}^{p}

This seems like the way to go but am stuck. Any suggestions? Thanks.

 

[dimension10]

Is there a way to do this without differentiation?

\left(a+b\right)^{p} \leq a^{p}+b^{p}


0<p<1 and a,b\geq 0

pulling the a out of the the first part and dividing by it to get

\left(1+\frac{b}{a}\right)^{p}\leq 1+\frac{b}{a}^{p}

This seems like the way to go but am stuck. Any suggestions? Thanks.

Use the binomial theorem?

 

[autobot.d]

p is not an integer though. Not sure binomial thm would work.

0 < p < 1

 

[dimension10]

p is not an integer though. Not sure binomial thm would work.

0 < p < 1

It works.

 

[autobot.d]

Did not know that, will do some research.

 

[dimension10]

Did not know that, will do some research.

It is Newton's generalisation that works. It is there on wikipedia.