$$L(p)=8\int_0^{2^{-\frac{1}{p}}}(1+(x^{-p}-1)^{1-p})^{\frac{1}{p}}dx, p\in \mathbb{R}, p\ge 1$$
May I ask if this $L$ function is increasing ot decreasing?
2026-03-26 01:07:48.1774487268
The monotone of an integral function
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Given $$L(p)=8\int_0^{2^{-\frac{1}{p}}}(1+(x^{-p}-1)^{1-p})^{\frac{1}{p}}dx, p\in \mathbb{R}, p\ge 1$$
Calculate $L'(p)$ using Leibnitz theorem for integrals
$$L'(p)=8\left( 1+(2-1)^{1-p} \right)^{\frac{1}{p}}\cdot2^{-\frac{1}{p}}\frac{\log 2}{p^2}-0=8\frac{\log 2}{p^2}$$
Can you solve the question from here?