Question Number 220020 by hardmath last updated on 04/May/25
![If f:[a,b]→[−1,∞) a,b∈R a ≤ b f-continuous Then prove that: (∫_a ^( b) (1+f(x))dx)^3 ≥ (b−a)^3 + 3(b−a)^2 ∫_a ^( b) f(x)dx](https://www.tinkutara.com/question/Q220020.png)
$$\mathrm{If}\:\:\:\mathrm{f}:\left[\mathrm{a},\mathrm{b}\right]\rightarrow\left[−\mathrm{1},\infty\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{a},\mathrm{b}\in\mathbb{R} \\ $$$$\:\:\:\:\:\:\:\mathrm{a}\:\leqslant\:\mathrm{b} \\ $$$$\:\:\:\:\:\:\:\mathrm{f}-\mathrm{continuous} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\left(\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\left(\mathrm{1}+\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}\right)^{\mathrm{3}} \geqslant\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{3}} +\:\mathrm{3}\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$
Answered by MrGaster last updated on 04/May/25
Commented by MrGaster last updated on 04/May/25
Solution (1)
Answered by MrGaster last updated on 04/May/25
Commented by MrGaster last updated on 04/May/25
Solution（2）
Commented by hardmath last updated on 04/May/25
$$ \\ $$One of the perfect solutions as always, thank you very much, my precious magical mathematician, you are great