Question

For $0\le p\le 1$ and for any positive $a,b$ let $I(p)=(a+b{)}^p,J(p)=a^p+b^p$, then

• A. $I(p)>J(p)$
• B. $I(p)\le J(p)$
• C. $I(p)<J(p)$ in $[0,\frac{P}{2}]$ & $I(p)>J(p)$ in $[\frac{P}{2},\infty )$
• D. $I(p)<J(p)$ in $[\frac{P}{2},\infty )$ & $J(p)>I(p)$ in $[0,\frac{P}{2}]$

Answer 1

Answer: (B)

Here, let $p=\frac{1}{m}$
then ${\left(a^p+b^p\right)}^{1/p}={\left(a^{1/m}+b^{1/m}\right)}^m$
$=a+b+k,k\ge 0$
$\therefore a^p+b^p\ge (a+b{)}^p\ge (a+b)$
$\Rightarrow J(p)\ge I(p)$