Question Number 63389 by mathmax by abdo last updated on 03/Jul/19
![f function integrable on [a,b] is max ∫_a ^b f(x)dx =∫_a ^b maxf(x)dx? if not give a opposite example .](https://www.tinkutara.com/question/Q63389.png)
$${f}\:{function}\:{integrable}\:{on}\:\left[{a},{b}\right] \\ $$$${is}\:{max}\:\int_{{a}} ^{{b}} {f}\left({x}\right){dx}\:=\int_{{a}} ^{{b}} \:{maxf}\left({x}\right){dx}?\:\:{if}\:{not}\:{give}\:{a}\:{opposite}\:{example}\:. \\ $$
Commented by MJS last updated on 03/Jul/19
$$\mathrm{opposite}\:\mathrm{example} \\ $$$$\underset{{a}} {\overset{{b}} {\int}}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){dx}=\frac{\mathrm{1}}{\mathrm{3}}\left({a}^{\mathrm{3}} −{b}^{\mathrm{3}} \right)−\left({a}−{b}\right) \\ $$$$\mathrm{max}\:\left(\frac{\mathrm{1}}{\mathrm{3}}\left({a}^{\mathrm{3}} −{b}^{\mathrm{3}} \right)−\left({a}−{b}\right)\right)\:=\frac{\mathrm{4}}{\mathrm{3}}\:\mathrm{with}\:{a}=−\mathrm{1}\wedge{b}=\mathrm{1} \\ $$$$ \\ $$$$\mathrm{max}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\:=\mathrm{1}\:\mathrm{with}\:{x}=\mathrm{0} \\ $$$$\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\mathrm{1}{dx}=\mathrm{2} \\ $$$$\frac{\mathrm{4}}{\mathrm{3}}\neq\mathrm{2} \\ $$
Commented by mathmax by abdo last updated on 03/Jul/19
$${thanks}\:{sir}\:{mjs}. \\ $$