Question Number 218626 by Nicholas666 last updated on 13/Apr/25
$$ \\ $$$$\:{Prove}\:{that}\:{for}\:{all}\:{real}\:{numbers}\:{a}\:{and}\:{b} \\ $$$${with}\:{a}<{b},\:{the}\:{following}\:{inequality}\:{holds}; \\ $$$$\left(\int_{{a}} ^{{b}} \mathrm{1}\:{dx}\right)^{\mathrm{3}} \leqslant\:\left({b}−{a}\right)\left(\int_{{a}} ^{{b}} \left({x}−{a}+\mathrm{1}\right)^{\mathrm{2}} {dx}\right)\left(\int_{{a}\:} ^{{b}} \frac{\mathrm{1}}{\left({a}−{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}\right) \\ $$$$ \\ $$
Commented by Nicholas666 last updated on 14/Apr/25
$${for}\:{those}\:{of}\:{you}\:{who}\:{marked}\:{the}\: \\ $$$${problem}\:{I}\:{posted}\:{inRed}\:{it}'{s}\:{because}\: \\ $$$${You}\:{didn}'{t}\:{break}\:{down}\:{the}\:{math}\:{with}\: \\ $$$${in}\:{depth}\:{analysis}. \\ $$