Question

For $0 \leq p \leq 1$ 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. $I (p) \leq J (p)$
• 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 $\left(a^{p}+b^{p}\right)^{1 / p}=\left(a^{1 / m}+b^{1 / m}\right)^{m}$
$=a+b+k, k \geq 0$
$\therefore a^{p}+b^{p} \geq(a+b)^{p} \geq(a+b)$
$\Rightarrow J(p) \geq I(p)$