NMTC Inter Level Problem 1

Algebra Level 4

$2x^2+2y^2+5z^2-2xy-4yz-4x-2z+15$

The minimum value of the above expression for real parameters $x,y,z$ is?

15 18 10 25

2 solutions

Sean Ty
Aug 24, 2014

Completing the square, we get

$(x^{2}-4x+4)+(x^{2}-2xy+y^{2})+(y^{2}-4yz+4z^{2})+(z^{2}-2z+1)+10$

$(x-2)^{2}+(x-y)^{2}+(y-2z)^{2}+(z-1)^{2}+10$

Minimum value is $\boxed{10}$ and is achieved when $x=2, y=2, z=1$

Awesome use of completing the square! :D

Krishna Ar - 6 years, 9 months ago

Thank you, as my teacher said: "Before advancing to higher maths, master the basics first!" :D

Sean Ty - 6 years, 9 months ago

:D...so very true!

Krishna Ar - 6 years, 9 months ago

Nanayaranaraknas Vahdam
Aug 23, 2014

Hint: Use Partial Differentiation

What about a calcu-less solution? :P (I did using calculus only )

Krishna Ar - 6 years, 9 months ago

put x, y, z = 1 and we get

2 + 2 + 5 - 2 - 4 - 4 - 2 + 15 = 12

Therefore, the minimum will be 12 or <12, hence, the answer is 10!!

Kartik Sharma - 6 years, 9 months ago

Troll logic

Nanayaranaraknas Vahdam - 6 years, 9 months ago

You really solved that way? :-o Mind-Blown again

Krishna Ar - 6 years, 9 months ago

that's right

Jeremiah Jocson - 6 years, 7 months ago

I love this short and effective solution. Sometimes we have to think out of the box in order to solve things fast.

Tín Phạm Nguyễn - 6 years, 3 months ago

Complete the square :)

Sean Ty - 6 years, 9 months ago

:) . Uh-Oh! Once again, the technique I hate.. :P

Krishna Ar - 6 years, 9 months ago

@Krishna Ar – Awwww, I once hated AM-GM, but now I really like it :P

Sean Ty - 6 years, 9 months ago

@Sean Ty – LOL...that's how preferences change!

Krishna Ar - 6 years, 9 months ago

@Sean Ty could you write a solution? I am very interested by completing the square

Nanayaranaraknas Vahdam - 6 years, 9 months ago

@Nanayaranaraknas Vahdam – Sure. Still shorter than differentiating. :O

Sean Ty - 6 years, 9 months ago