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) \(\mathrm{I}(\mathrm{p})>\mathrm{J}(\mathrm{p})\)
(B) \(\mathrm{I}(\mathrm{p}) \leq \mathrm{J}(\mathrm{p})\)
(C) \(I(p)<J(p)\) in \(\left[0, \frac{p}{2}\right] \& I(p)>J(p)\) in \(\left[\frac{p}{2}, \infty\right)\)
(D) \(\mathrm{I}(\mathrm{p})<\mathrm{J}(\mathrm{p})\) in \(\left[\frac{\mathrm{P}}{2}, \infty\right) \& \mathrm{~J}(\mathrm{p})<\mathrm{I}(\mathrm{p})\) in \(\left[0, \frac{\mathrm{P}}{2}\right]\)

Answer 1

Ans: (B)
Hint : let \(p=\frac{1}{n}\), then \(\left(a^{p}+b^{p}\right)^{1 / p}=\left(a^{\frac{1}{n}}+b^{\frac{1}{n}}\right)^{n}=a+b+k, k \geq 0 \therefore a^{p}+b^{p} \geq(a+b)^{p} \geq a+b\)

Answer 2

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Answer 3

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