Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

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 .

$${f}\:{function}\:{integrable}\:{on}\:\left[{a},{b}\right] \\ $$$${is}\:{max}\:\int_{{a}} ^{{b}} {f}\left({x}\right){dx}\:=\int_{{a}} ^{{b}} \:{maxf}\left({x}\right){dx}?\:\:{if}\:{not}\:{give}\:{a}\:{opposite}\:{example}\:. \\ $$

Commented by MJS last updated on 03/Jul/19

opposite example  ∫_a ^b (1−x^2 )dx=(1/3)(a^3 −b^3 )−(a−b)  max ((1/3)(a^3 −b^3 )−(a−b)) =(4/3) with a=−1∧b=1    max (1−x^2 ) =1 with x=0  ∫_(−1) ^1 1dx=2  (4/3)≠2

$$\mathrm{opposite}\:\mathrm{example} \\ $$$$\underset{{a}} {\overset{{b}} {\int}}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){dx}=\frac{\mathrm{1}}{\mathrm{3}}\left({a}^{\mathrm{3}} −{b}^{\mathrm{3}} \right)−\left({a}−{b}\right) \\ $$$$\mathrm{max}\:\left(\frac{\mathrm{1}}{\mathrm{3}}\left({a}^{\mathrm{3}} −{b}^{\mathrm{3}} \right)−\left({a}−{b}\right)\right)\:=\frac{\mathrm{4}}{\mathrm{3}}\:\mathrm{with}\:{a}=−\mathrm{1}\wedge{b}=\mathrm{1} \\ $$$$ \\ $$$$\mathrm{max}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\:=\mathrm{1}\:\mathrm{with}\:{x}=\mathrm{0} \\ $$$$\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\mathrm{1}{dx}=\mathrm{2} \\ $$$$\frac{\mathrm{4}}{\mathrm{3}}\neq\mathrm{2} \\ $$

Commented by mathmax by abdo last updated on 03/Jul/19

thanks sir mjs.

$${thanks}\:{sir}\:{mjs}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com