If x, y and z are all non-zero numbers, what is the minimum value of x

Question

If x, y and z are all non-zero numbers, what is the minimum value of  x^4 + z^2 -4yz + 4y^2 -4x^2y + 2x^2z - 2  ?

A. -2
B. -1
C. 0
D. 1
E. 2

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IanStewart · Accepted Answer

The standard way to do a question like this is to find a factorization involving a square (and perhaps other terms), and then, by noticing that squares can never be less than zero in value, finding the minimum or maximum value of the expression by making the square exactly equal to zero. yashikaaggarwal has done that perfectly above, so I'll just point out a different way one could solve almost instantly. If you notice that when x, y and z are all exactly equal to zero, the value of the expression is -2, then it must be true that when x, y and z are infinitesimally different from zero (so are all equal, say, to 0.00000001) the value of the expression will be negligibly different from -2. So its minimum value clearly must be less than -1, and only answer A can be right.

yashikaaggarwal · Answer

X^4+Z^2+4Y^2-4YZ-4X^2Y+2X^2Z-2
Compare it with (a+b+c)^2
Where A = -x^2
B = 2y
C = -Z
{X^4+Z^2+4Y^2-4YZ-4X^2Y+2X^2Z}-2
=> (-X^2+2y-Z)^2-2
For minimum value put (-X^2+2y-Z)^2 = 0
=> (-X^2+2y-Z)^2 = 0
=> 0-2 = -2

IMO A

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Kinshook · Answer

Asked: If x, y and z are all non-zero numbers, what is the minimum value of  x^4 + z^2 -4yz + 4y^2 -4x^2y + 2x^2z - 2  ?

z^2 - 4yz + 4y^2 + x^2 (x^2 - 4y + 2z) - 2
(z - 2y)^2 + x^4 + 2x^2(z - 2y) - 2
((z - 2y) + x^2)^2 - 2

Min value = - 2
Since min ((z - 2y) + x^2)^2 = 0 ; when z = 2y & x = 0

IMO A

Lipun · Answer

x^4 + z^2 - 4yz + 4y^2 - 4x^2y + 2x^2z - 2

=    - 2

=  (x^2 - 2y +z)^2 - 2

Since the first term is a square, therefore, the minimum value can only be zero.
=> Min value of the expression = -2

saumyagupta1602 · Answer

min val x^4= 1 for x=+1 or -1 expression has only x^2 so x=1 or -1 doesn't matter
for z^2=1 z=1 or -1 
taking x^4, x^2 and z^2=1 we get 1+1 -4yz + 4y^2 -4y+2z-2
checking for z=-1; 4y^2-8y+2 so case 1 y=1 ie 4-8+2=-2 or case 2 y=-1 ie 4+8+2=14 , min so y=1 and -2
checking for z=1 ; 4y^2-2 so y=+1or -1 gives Y^2=1 and val= 2
min(-2,2)=-2 answer

ShankSouljaBoi · Answer

Yu have taken x= 0 ; however, it is given in the stem that x y and z cant assume zero values.

BlueRocketAsh · Answer

x^4+z^2−4yz+4y^2−4x^2y+2x^2z−2
= (x^4+z^2+4y^2−4x^2y+2x^2z−4yz)−2
= (x^2 - 2y + z)^2 - 2
Here all the variable are inside a square term. Minimum value of the square term is 0
Therefore minimum value of the expression is -2

Ruddy370 · Answer

I just assumed 1 as the value for all x y z, since taking any negative would lead to a + sign in a few terms. Am I missing something?

egmat · Answer

Hi Ruddy370,

Good news first: your value of  -2  is exactly right. But your reasoning got you there partly by luck, and that's the gap worth closing.

The problem with pure substitution:  plugging in one set of values (here x = y = z =  1 ) tells you  a  value the expression can take - it does  not  prove that value is the smallest possible. To claim a minimum, you have to show that  no other  choice of x, y, z can push it lower. Trying one triple can't do that on its own.

You happened to land on the true minimum because at x = y = z =  1  something special quietly happens. Look at the factorization the others posted:

x^4 + z^2 - 4yz + 4y^2 - 4x^2y + 2x^2z - 2 = (x^2 - 2y + z)^2 - 2

A square is  always >= 0 , so the whole thing is smallest exactly when that square equals  0 , leaving  -2 . Now check your triple: at x = y = z =  1 , the inside is  1  -  2  +  1  =  0 . That's why you got  -2  - you accidentally picked values that zero out the square. Pick almost any other triple and the square is positive, so the expression is  bigger  than  -2 , never smaller.

So your instinct "negatives just add positive terms" isn't a reliable rule - the real reason  -2  is the floor is the perfect square.

Quick check to feel the difference: 
- What's the minimum of (x -  3 )^2 -  2 ?
- If you'd plugged in x =  1 , you'd get  2 . Plug in x =  3 , you get  -2 .

Same lesson: substitution only finds the minimum if you happen to hit the value that zeroes the square. The square form  guarantees  it.

Answer: A

If x, y and z are all non-zero numbers, what is the minimum value of x

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GMAT Problem Solving (PS) Questions

If x, y and z are all non-zero numbers, what is the minimum value of x

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards