1. Optimization Problem

Hi, I had a problem that I just had no idea how to do. I was told it was an optimization but I am still confused. The problem states :
Let S denote the set of points with the coordinates (x,y) such that 0≤x≤15 and y^2 = 150- 10x. Find the coordinates of the point(s) of S nearest to the origin. Also find the coordinates of the point(s) of S farthest from the origin. Justify your answer.
All i have gotten is y= squareroot(150-10x) and that is not much.

2. Originally Posted by calc_help123
Hi, I had a problem that I just had no idea how to do. I was told it was an optimization but I am still confused. The problem states :
Let S denote the set of points with the coordinates (x,y) such that 0≤x≤15 and y^2 = 150- 10x. Find the coordinates of the point(s) of S nearest to the origin. Also find the coordinates of the point(s) of S farthest from the origin. Justify your answer.
All i have gotten is y= squareroot(150-10x) and that is not much.
remember the distance formula ? distance of the point (x,y) from the origin is ...

$d = \sqrt{(x - 0)^2 + (y - 0)^2}$

$d = \sqrt{x^2 + y^2}$

substitute for $y^2$

$d = \sqrt{x^2 - 10x + 150}$

let $d = \sqrt{z}$

if z is maximized/minimized, so is d ...

$z = x^2 - 10x + 150$

$\frac{dz}{dx} = 2x - 10$

$2x - 10 = 0$

$x = 5$

$\frac{d^2z}{dz^2} = 2 > 0$

$x = 5$ will yield a minimum distance

the endpoints, $x = 0$ and $x = 15$ , will be maximums ... calculate which yields the absolute maximum distance.