Question Number 129873 by Bird last updated on 20/Jan/21
$${let}\:{m}={inff}\left({x}\right)_{{x}\in\left[{a},{b}\right]} \\ $$$${and}\:{M}={supf}\left({x}\right)_{{x}\in\left[{a},{b}\right]} \\ $$$${prove}\:{that}\:\left({b}−{a}\right)^{\mathrm{2}} \leqslant\int_{{a}} ^{{b}} {f}\left({x}\right){dx}.\int_{{a}} ^{{b}} \:\frac{{dx}}{{f}\left({x}\right)} \\ $$$$\leqslant\left({b}−{a}\right)^{\mathrm{2}} ×\frac{\left({m}+{M}\right)^{\mathrm{2}} }{\mathrm{4}{mM}} \\ $$