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Question Number 129873 by Bird last updated on 20/Jan/21
![let m=inff(x)_(x∈[a,b]) and M=supf(x)_(x∈[a,b]) prove that (b−a)^2 ≤∫_a ^b f(x)dx.∫_a ^b (dx/(f(x))) ≤(b−a)^2 ×(((m+M)^2 )/(4mM))](Q129873.png)
$${let}\:{m}={inff}\left({x}\right)_{{x}\in\left[{a},{b}\right]} \\ $$$${and}\:{M}={supf}\left({x}\right)_{{x}\in\left[{a},{b}\right]} \\ $$$${prove}\:{that}\:\left({b}−{a}\right)^{\mathrm{2}} \leqslant\int_{{a}} ^{{b}} {f}\left({x}\right){dx}.\int_{{a}} ^{{b}} \:\frac{{dx}}{{f}\left({x}\right)} \\ $$$$\leqslant\left({b}−{a}\right)^{\mathrm{2}} ×\frac{\left({m}+{M}\right)^{\mathrm{2}} }{\mathrm{4}{mM}} \\ $$