WBJEE · Maths · Basic of Mathematics
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 \(l(p)>J(p)\)
- B \(l(p) \leq J(p)\)
- C \(l(p) < J{(} p)\) in \(\left[0 ,\frac{p}{2}\right]\) and \(l(p)>J(p)\) in \(\left[\frac{p}{2}, \infty\right)\)
- D \(l(p) < J(p)\) in \(\left[\frac{p}{2}, \infty\right)\) and \(J(p) < l(p)\) in \(\left[0, \frac{p}{2}\right]\)
Answer & Solution
Correct Answer
(B) \(l(p) \leq J(p)\)
Step-by-step Solution
Detailed explanation
Here, let \(p=\frac{1}{m}\) then \(\left(a^{p}+b^{p}\right)^{1 / p}=\left(a^{1 / m}+b^{1 / m}\right)^{m}\) \(=a+b+k, k \geq 0\) \(\therefore \quad a^{p}+b^{p} \geq(a+b)^{p} \geq(a+b)\) \(\Rightarrow \quad J(p) \geq I(p)\)
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