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Question Number 42133 by rahul 19 last updated on 18/Aug/18

$$\mathrm{Let}\mathrm{a},\mathrm{b},\mathrm{c}ϵ\mathrm{R}\mathrm{such}\mathrm{that}\mathrm{a}+2\mathrm{b}+\mathrm{c}=4.$$$$\mathrm{Find}\mathrm{maximum}\mathrm{value}\mathrm{of}(\mathrm{ab}+\mathrm{bc}+\mathrm{ca}).$$

Answered by MrW3 last updated on 18/Aug/18

$$b=\frac{4-a-c}{2}=2-\frac{a+c}{2}$$$$F(a,c)=ab+bc+ca=(a+c)b+ac=2(a+c)-\frac{{(a+c)}^2}{2}+ac$$$$\frac{\partial F}{\partial a}=2-(a+c)+c=2-a=0\Rightarrow a=2$$$$\frac{\partial F}{\partial c}=2-(a+c)+a=2-c=0\Rightarrow c=2$$$$max.F=F(2,2)=2\times 4-\frac{16}{2}+2\times 2=4$$

Commented by rahul 19 last updated on 18/Aug/18

Ausgezeichnet! ����

Commented by MrW3 last updated on 18/Aug/18

Danke sehr!

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