f-integrable-continue-on-a-b-let-m-inf-f-x-and-M-sup-f-x-x-a-b-prove-that-b-a-2-a-b-f-x-dx-a-b-dx-f-x-b-a-2-4-m-M-2-mM-

Question Number 107283 by mathmax by abdo last updated on 09/Aug/20

f integrable continue on [a, b] let m = \inf f (x) and M = \sup f (x)

(x \in [a, b] prove that {(b - a)}^{2} ⩽ \int_{a}^{b} f (x) dx \times \int_{a}^{b} \frac{dx}{f (x)} ⩽ \frac{{(b - a)}^{2}}{4} \frac{{(m + M)}^{2}}{mM}