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If a+2b+c=4,then what is the maximum value of ab+bc+ac?

If a+2b+c=4,then what is the maximum value of ab+bc+ac?

Grade:11

2 Answers

vinayak prabhu sd
23 Points
7 years ago
First,
c=4-a-2b......(1),then
substute it in ab+bc+ca.........(2)
now differentiate this equation 2 times first wrt a and then wrt b.
now that you have two equations, solve them using linear equations in two variables and get values for a and b.since you got those answers after differentiating them, you have their greatest values. now get the value for c using eqn (1).now again substute all the values in (2) and you thus have its greatest value.
jagdish singh singh
173 Points
7 years ago
\hspace{-0.7 cm}$Using $(A+B)^2\geq 4AB$ and equality hold when $A=B$\\\\ So $(a+b+b+c)^2\geq 4(a+b)(b+c)$\\\\ So $4(ab+bc+ca+b^2)\leq (a+2b+c)^2=4^2=16$\\\\ So $ab+bc+ca\leq 4-b^2\leq 4$\\\\ So $\max(ab+bc+ca) = 4$ when $a=c=2$ and $b=0$

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