Let $l$ be the minimum value of $2(y-2)^2 +4(x -7)^2 +(y+4)^2$. Find $[2018/l]$ where $[\cdot]$ is the greatest integer function.

Question

$l$ is equal to the minimum value of the expression $2(y-2)^2 + 4(x -7)^2 + (y+4)^2$. Find $[2018/l]$ where $[\cdot]$ denotes the greatest integer function.

Here is my approach:

In order to obtain the minimum value of the expression, one of the terms has to be zero, so let $x=7$ and $y=2$. We get $(2+4)^2=36$ and $[2018/36]=56$.

Is this answer right?

Answer 1

Guide:

Nope, that is not the right way.

Letting $x=7$ is a good move.

now, let's focus on the $y$ terms. To minimize

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

You can differentiate the function with respect to $y$ and equate it to zero to solve for the $y$ corresponding to the minimum value.

Note that the function is convex hence there is a minimum.