What point on the surface z is closest to the origin $z = x ^ { 2 } + y - 5$
Hint : Minimize the square of the distance from $( x , y , z )$ to the origin.

A) $( 2.5,0 , - 2.5 )$
B) $\left( - 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ and $\left( 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
C) $( 0,2.5 , - 2.5 )$
D) $\left( 0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
E) $( 0,2.5 , - 2.5 )$ and $( 0,2.5,2.5 )$

Question

What point on the surface z is closest to the origin    $z = x ^ { 2 } + y - 5$  
Hint : Minimize the square of the distance from $( x , y , z )$ to the origin.
 
A) $( 2.5,0 , - 2.5 )$
B) $\left( - 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ and $\left( 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
C) $( 0,2.5 , - 2.5 )$
D) $\left( 0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
E) $( 0,2.5 , - 2.5 )$ and $( 0,2.5,2.5 )$

Quizplus · Accepted Answer

To find the point on the surface closest to the origin, we minimize the distance squared: $D^2 = x^2 + y^2 + (x^2 + y - 5)^2$. Solving the system of equations derived from setting the partial derivatives of $D^2$ with respect to $x$ and $y$ to zero gives the points $\left( - 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ and $\left( 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$.

What Point on the Surface Z Is Closest to the Origin