This question aims to learn the basic methodology for optimizing a mathematical function (maximizing or minimizing).
Critical points are the points where the value of a function is either maximum or minimum. To calculate the critical point(s), we equate the first derivative’s value to 0 and solve for the independent variable. We can use the second derivative test to find maxima/minima. For the given question, we can minimize the distance function of the desired point from the origin as explained in the below answer.
Expert Answer
Given:
\[ y^{ 2 } \ = \ 9 \ + \ x \ z \]
Let $ ( x, \ y, \ z ) $ be the point that is nearest to the origin. The distance of this point from the origin is calculated by:
\[ d = \sqrt{ x^{ 2 } + y^{ 2 } + z^{ 2 } } \]
\[ \Rightarrow d^{ 2 } = x^{ 2 } + y^{ 2 } + z^{ 2 } \]
\[ \Rightarrow d^{ 2 } = x^{ 2 } + 9 + x z + z^{ 2 } \]
To find this point, we simply need to minimize this $ f(x, \ y, \ z) \ = \ d^{ 2 } $ function. Calculating the first derivatives:
\[ f_x = 2x + z \]
\[ f_z = x + 2z \]
Finding critical points by putting $ f_x $ and $ f_z $ equal to zero:
\[ 2x + z = 0\]
\[ x + 2z = 0\]
Solving the above system yields:
\[ x = 0\]
\[ z = 0\]
Consequently:
\[ y^{ 2 } = 9 + xz = 9 + (0)(0) = 0 \]
\[ \Rightarrow = y = \pm 3 \]
Hence, the two possible critical points are $ (0, 3, 0) $ and $ (0, -3, 0) $. Finding the second derivatives:
\[ f_{xx} = 2 \]
\[ f_{zz} = 2 \]
\[ f_{xz} = 1 \]
\[ f_{zx} = 1 \]
Since all second derivatives are positive, the calculated critical points are at a minimum.
Numerical Result
Points Closest to the origin = $ (0, 0, 5)$ and $ (0, 0, -5) $
Example
Find the points on the surface $ z^2 = 25 + xy $ nearest to the origin.
Here, the distance function becomes:
\[ d = \sqrt{ x^{ 2 } + y^{ 2 } + z^{ 2 } } \]
\[ \Rightarrow d^{ 2 } = x^{ 2 } + y^{ 2 } + z^{ 2 } \]
\[ \Rightarrow d^{ 2 } = x^{ 2 } + y^{ 2 } + 25 + xy \]
Calculating first derivatives and equating to zero:
\[ f_x = 2x + y \Rightarrow 2x + y = 0\]
\[ f_y = x + 2y \Rightarrow x + 2y = 0\]
Solving the above system yields:
\[ x = 0 \text{and} y = 0\]
Consequently:
\[ z^{ 2 } = 25 + xy = 25 \]
\[ \Rightarrow = z = \pm 5 \]
Hence, the two possible critical points are $ (0, 3, 0) $ and $ (0, -3, 0) $. Finding the second derivatives:
\[ f_{xx} = 2 \]
\[ f_{yy} = 2 \]
\[ f_{xy} = 1 \]
\[ f_{yx} = 1 \]
Since all second derivatives are positive, the calculated critical points are at a minimum.
Points Closest to the origin = $ (0, 0, 5) $ and $ (0, 0, -5) $