Find out the answer to "Find the point(s) on the surface at which..."

High School
Algebra
Pre-Algebra
Negative numbers and coordinate plane

Phoebe

Answered question

2021-10-31

Find the point(s) on the surface at which the tangent plane is horizontal.

z = x y + \frac{1}{x} + \frac{1}{y}

Answer & Explanation

ensojadasH

Skilled2021-11-01Added 100 answers

Set the partial derivatives with respect to x and y to zero after finding them. To find y, first solve for x in the partial with respect to y. Then, put the answer back into the partial with regard to y and back into the partial with respect to x. Since the denominator cannot be 0, y must equal 1. To find x, plug 1 into the x equation.
$z = x y + \frac{1}{x} + \frac{1}{y}$
$f_{x} (x, y) = y - \frac{1}{x^{2}} = 0$
$f_{y} (x, y) = x - \frac{1}{y^{2}} = 0$
$x = \frac{1}{y^{2}}$
$y - \frac{1}{\frac{1}{y^{2}}} =$
$- y^{2} + y = 0$
$y (- y + 1) = 0$
$y = 1$
$x = \frac{1}{1^{2}} = 1$
To determine the third coordinate, enter the point (1,1) into z.
$z (1, 1) = 1 \cdot 1 + \frac{1}{1} + \frac{1}{1} = 3$
Result: $(x, y, z) = (1, 1, 3)$

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