Problem Solving

navami wrote:3 it is

Rahul@gurome wrote:

thp510 wrote:...
Not getting it. What does the equation look like before you get to 4x=4 or 6y=12 using the differentiation rule. I'm still stuck at the following:

2(x^2-2x)+3(y^2-4y)+18

To use the differentiation rule, you need to know differential calculus which is not a part of GMAT. You can proceed as I mentioned earlier. Let's do it again. To minimize the expression, we have to minimize each term of it. Simple algebraical method to do it is to rearrange the terms in such a way that we get square terms. This is because minimum value of a square term is zero.

# 2x² + 3y² - 4x - 12y + 18
= 2(x² - 2x) + 3(y² - 4y) + 18 .................... What you've done
= 2(x² - 2x + 1) + 3(y² - 4y + 4) + 4
= 2(x - 1)² + 3(y - 2)² + 4

For minimum value of the expression, 2(x - 1)² and 3(y - 2)² must be minimum, i.e. equal to zero. Thus minimum value of the expression is 4.

GMATGuruNY wrote:

winnerhere wrote:Find the minimum value of an expression

2 (x^2) + 3 (y^2) - 4x - 12y + 18

1) 18

2)10

3) 4

4) 0

5) -10

I received a PM asking me to comment. Let's start by reorganizing the expression.

2 (x^2) + 3 (y^2) - 4x - 12y + 18

= 2x^2 - 4x + 3y^2 - 12y + 18

= 2(x^2 - 2x) + 3(y^2 - 4y) + 18

Let's handle each part of the expression separately, starting with (x^2 - 2x).
We need to complete the square inside the (). We're looking for a value that when added to itself will yield a sum of -2 (the coefficient of -2x).
Thus, the needed value inside the parentheses is -1:

(x-1)^2 = x^2 - 2x + 1

Now in the expression 2(x^2 - 2x) + 3(y^2 - 4y) + 18 we replace (x^2 - 2x) with (x^2 - 2x +1):

2(x^2 - 2x + 1) + 3(y^2 - 4y) + 18

But we can't change the value of the expression. Since we placed +1 inside the (), and the entire expression inside the () is being multiplied by 2, we need to subtract 2*1 = 2 from the whole expression so that its value is unchanged:

2(x^2 - 2x + 1) + 3(y^2 - 4y) + 18 - 2

= 2(x-1)^2 + 3(y^2 - 4y) + 16

Now let's handle the second part of the expression, (y^2 - 4y).

We need to complete the square inside the (). We're looking for a value that when added to itself will yield a sum of -4 (the coefficient of -4y). Thus, the needed value inside the parentheses is -2:

(y-2)^2 = y^2 - 4y + 4

Now in the expression 2(x-1)^2 + 3(y^2 - 4y) + 16 we replace (y^2 - 4y) with (y^2 - 4y + 4):

2(x-1)^2 + 3(y^2 - 4y + 4) + 16

But we can't change the value of the expression. Since we placed +4 inside the (), and the entire expression inside the () is being multiplied by 3, we need to subtract 3*4 = 12 from the whole expression so that the value of the expression is unchanged:

2(x-1)^2 + 3(y^2 - 4y + 4) + 16 - 12

= 2(x-1)^2 + 3(y-2)^2 + 4.

Now we can determine the minimum value of the expression. The expression will be minimized when x=1 and y=2 so that the first two terms equal 0:

2(x-1)^2 + 3(y-2)^2 + 4 = 2(1-1)^2 + 3(2-2)^2 + 4 = 0 + 0 + 4 = 4.

The correct answer is C.

Hope this helps!