How-do-we-prove-the-area-of-a-square-rectangle-is-at-its-maximum-when-both-sides-are-equal-So-if-we-have-a-rectangle-of-sides-a-and-b-how-do-we-show-its-maximized-when-a-b-

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Question Number 5414 by FilupSmith last updated on 14/May/16

$$\mathrm{How}\mathrm{do}\mathrm{we}\mathrm{prove}\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{a}\mathrm{square}/\mathrm{rectangle}$$$$\mathrm{is}\mathrm{at}\mathrm{its}\mathrm{maximum}\mathrm{when}\mathrm{both}\mathrm{sides}\mathrm{are}\mathrm{equal}.$$$$$$$$\mathrm{So},\mathrm{if}\mathrm{we}\mathrm{have}\mathrm{a}\mathrm{rectangle}\mathrm{of}\mathrm{sides}a\mathrm{and}b,$$$$\mathrm{how}\mathrm{do}\mathrm{we}\mathrm{show}\mathrm{its}\mathrm{maximized}\mathrm{when}a=b?$$

Answered by 123456 last updated on 14/May/16

$$\mathrm{lets}\mathrm{a}\mathrm{retangle}\mathrm{with}\mathrm{side}a\mathrm{and}b$$$$\mathrm{lets}s\mathrm{be}\mathrm{area}\mathrm{and}p\mathrm{the}\mathrm{perimeter}$$$$p=2a+2b$$$$s=ab$$$$\mathrm{lets}\mathrm{get}\mathrm{all}\mathrm{rectangle}\mathrm{with}\mathrm{constant}$$$$\mathrm{perimeter},\mathrm{we}\mathrm{have}$$$$a=\frac{p}{2}-b0⩽b⩽p/2$$$$s=ab$$$$=(\frac{p}{2}-b)b$$$$=-b^2+\frac{pb}{2}$$$$\mathrm{finding}\mathrm{critic}\mathrm{point}$$$$s_b=-2b+\frac{p}{2}$$$$s_b=0\iff b=\frac{p}{4}\iff a=\frac{p}{4}$$$$s_{bb}=-2<0$$$$\mathrm{so}b=2\mathrm{is}\mathrm{a}\mathrm{point}\mathrm{of}\max$$$$\mathrm{so}$$$$“\mathrm{in}\mathrm{the}\mathrm{conjunt}\mathrm{of}\mathrm{rectangle}\mathrm{with}\mathbf{same}$$$$\mathbf{perimeter},\mathrm{the}\mathrm{square}\mathrm{have}\mathrm{the}\mathrm{greatest}$$$${\mathrm{area}}^{″}$$

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