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-

Question Number 129873 by Bird last updated on 20/Jan/21

l e t m = i n f f {(x)}_{x \in [a, b]}

a n d M = s u p f {(x)}_{x \in [a, b]}

p r o v e t h a t {(b - a)}^{2} ⩽ \int_{a}^{b} f (x) d x . \int_{a}^{b} \frac{d x}{f (x)}

⩽ {(b - a)}^{2} \times \frac{{(m + M)}^{2}}{4 m M}