Question

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

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

Answer 1

The correct option is B $I\left(p\right)\le J\left(p\right)$

Let $a=3,b=4$

When $p=0.5$
$I\left(P\right)=5;J\left(P\right)=7$
$J\left(P\right)>I\left(P\right)$
Now $a=\frac{1}{3};b=\frac{1}{4}$

When $P=1$
$I\left(P\right)=\frac{7}{12};J\left(P\right)=\frac{7}{12}$
$I\left(p\right)=J\left(p\right)$

$I\left(p\right)\le J\left(p\right)$