Non-Homogenous Inequality

Algebra Level 4

$\large x+y+z+xy+yz+xz$

Over all real $x,y$ and $z$ satisfying $x^2+y^2+z^2 = 1$ , find the minimum value of the above expression.

Inspiration .

The answer is -1.

1 solution

Manuel Kahayon
Mar 11, 2016

Let $x+y+z=N$

So, $N^2= (x^2+y^2+z^2)+2(xy+yz+xz) = 1+2(xy+yz+xz)$ (Since $x^2+y^2+z^2 = 1$ )

Giving us $xy+yz+xz = \frac{N^2-1}{2}$

So, our original equation is $x+y+z+xy+yz+xz$

= $N+\frac{N^2-1}{2} = \frac{N^2+2N-1}{2}$

By the formula for the minimum value of a quadratic formula, we get the minimum value to be $\boxed{-1}$

The minimum value is attained when $x+y+z = -1$ and $x^2+y^2+z^2 = 1$

I won't bother to prove that there exist $x,y$ and $z$ which satisfy the equation, I'll just state that $(x,y,z) = (-\frac{2}{3}, -\frac{2}{3}, \frac{1}{3})$ satisfy the equations.

Same solution, thanks for being inspired by me

P C - 5 years, 3 months ago

Not really, this problem was also inspired by you!

Manuel Kahayon - 5 years, 3 months ago

Finally figured out how to work out links in Brilliant.org.

Manuel Kahayon - 5 years, 3 months ago

@Manuel Kahayon – Good for you! For future reference, here is how a bunch of other things can be done :)

Calvin Lin Staff - 5 years, 3 months ago

I am sorry. But; I don't get this. What does N =?

Fredrick Prueter - 5 years, 2 months ago

$N= x+y+z$

Manuel Kahayon - 5 years, 2 months ago

I see it now. I didn't see the lengthy steps to take to get to your solution until now. Thank you!

Fredrick Prueter - 5 years, 2 months ago

This is why it is important to keep consistent notation throughout the solution. Your first line previously stated $x+y + z = n$ , and I'm switched it out to $N$ .

Calvin Lin Staff - 5 years, 2 months ago

@Calvin Lin – Oh, sorry, I did not notice that.... Thanks a bunch, sir Calvin!

Manuel Kahayon - 5 years, 2 months ago

Same solution!!

Aakash Khandelwal - 5 years, 3 months ago

Non-Homogenous Inequality

The answer is -1.

1 solution

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