determine min value of ab+bc+ac or a^2+b^2+c^2=1

determine min value of ab+bc+ac or a^2+b^2+c^2=1

Help please

StevenTing said:

determine min value of ab+bc+ac or a^2+b^2+c^2=1

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I'm sorry, but I don't understand these two exercises...?

For the first one, you are supposed to find the minimum possible value of ab + bc + ac, but this could be anything, because they've given you no limitations on the values of the three variables.

For the second one, you are given that the value of a[sup:2mq8r6jq]2[/sup:2mq8r6jq] + b[sup:2mq8r6jq]2[/sup:2mq8r6jq] + c[sup:2mq8r6jq]2[/sup:2mq8r6jq] is one, so how would one "minimize" this?

Please consult with your instructor regarding clarification of the instructions for each of these two exercises. Thank you!

Eliz.

Hello, StevenTing!

I believe you have misstated the problem.
It looks suspiciously like a minimization problem with a constraint.

\(\displaystyle \text{Determine minimum value of: }\:xy+yz + xz\:\text{ subject to: }x^2+y^2+z^2\:=\:1\)

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If I am correct, I recommend Lagrange Multipliers . . .

\(\displaystyle F(x,y,z,\lambda) \;=\;xy + yz + xz + \lambda(x^2+y^2+z^2-1)\)


Find the four partial derivatives and equate to zero . . .

. . \(\displaystyle \begin{array}{ccccc}f_x &=&y + z + 2\lambda x &=& 0 \\ \\[-3mm] f_y & =& x + z + 2\lambda y &=& 0 \\ \\[-3mm] f_z &=& y + x + 2\lambda z &=& 0 \\ \\[-3mm] f_{\lambda} &=& x^2+y^2+z^2-1 &=& 0 \end{array}\)


\(\displaystyle \text{Then solve the system for }x,y,z.\)