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Algebra Level 5

If $a^2+b^2+c^2=1$ , find the sum of maximum and minimum value of $ab+bc+ca$ .

1 solution

Tanishq Varshney
Nov 25, 2015

let $ab+bc+ca=t$

$(a+b+c)^2=1+2t$

as $a^2+b^2+c^2=1$

$\rightarrow 1+2t \geq 0$

$t \geq -\frac{1}{2}$

Also $\large{(a-b)^2+(b-c)^2+(c-a)^2=2(a^2+b^2+c^2)-2t}$

$\rightarrow 2-2t \geq 0$

$t \leq 1$

Thus sum of minimum and maximum value is $1-1/2=0.5$

I found the minimum value by your method only but found the minimum using Cauchy-Schwarz Inequality.

Kushagra Sahni - 5 years, 6 months ago

Same way bro

Department 8 - 5 years, 6 months ago

Only off by 0.03 :"v

Nanda Rahsyad - 5 years, 6 months ago

I did same!!

Dev Sharma - 5 years, 6 months ago

We can also use am,gm,gm inequality

Rohan K - 5 years, 6 months ago

how to do by using am gm hm inequality

n n - 5 years, 6 months ago