How to prove an L$^p$ type inequality Let $a,b\in[0,\infty)$ and let $p\in[1,\infty)$. How can I prove$$a^p+b^p\le(a^2+b^2)^{p/2}.$$

Some hints: 


*

*By homogeneity, we can assume that $b=1$. 

*Let $f(t):=(t^2+1)^{p/2}-t^p-1$ for $t\geq 0$. We have $f'(t)=p((t^2+1)^{p/2-1}-t^{p-1})$. We have $t^2+1\geq t^2$, so the derivative is non-negative/non-positive if $p\geq 2$ or $p<2$. 

*Deduce the wanted inequality (when it is reversed or not). 

