Find and classify the critical points of z = (22 = 4x) (y² 6y)Local maximums: DNELocal minimums: DNESaddle points: DNEFor each classification, enter a list of ordered pairs (x, y) where the max/miare no points for a classification, enter DNE.Check Answer

Answered: Find and classify the critical points…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find and classify the critical points of z = (22 = 4x) (y² 6y)
Local maximums: DNE
Local minimums: DNE
Saddle points: DNE
For each classification, enter a list of ordered pairs (x, y) where the max/mi
are no points for a classification, enter DNE.
Check Answer

Expert Solution

Step 1

Given function is $z = (x^{2} - 4 x) (y^{2} - 6 y)$

Let $z = f (x, y)$

Hence, $f (x, y) = (x^{2} - 4 x) (y^{2} - 6 y)$

Differentiate partially $f (x, y) = (x^{2} - 4 x) (y^{2} - 6 y)$ with respect to $x$

$\begin{matrix} \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} ((x^{2} - 4 x) (y^{2} - 6 y)) \\ = (2 x - 4) (y^{2} - 6 y) \end{matrix}$

Hence, $\frac{\partial f}{\partial x} = (2 x - 4) (y^{2} - 6 y)$

Differentiate partially $f (x, y) = (x^{2} - 4 x) (y^{2} - 6 y)$ with respect to $y$

$\begin{matrix} \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} ((x^{2} - 4 x) (y^{2} - 6 y)) \\ = (x^{2} - 4 x) (2 y - 6) \end{matrix}$

Hence, $\frac{\partial f}{\partial y} = (x^{2} - 4 x) (2 y - 6)$

Now, for critical point equate $\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} = 0$

Hence, $(2 x - 4) (y^{2} - 6 y) = 0....... (1)$

And $(x^{2} - 4 x) (2 y - 6) = 0....... (2)$

Since, $(2 x - 4) (y^{2} - 6 y) = 0$

$\begin{array}{l} \Rightarrow (2 x - 4) (y^{2} - 6 y) = 0 \\ \Rightarrow x = 2, y = 0 and y = 6 \end{array}$

Substitute $x = 2$ in equation (2) we get

$y = 3$

Hence, critical point is $(2, 3)$ .

Substitute $y = 0$ in equation (2) we get

$x = 0 a n d x = 4$

Hence, critical point is $(0, 0) a n d (4, 0)$ .

Substitute $y = 6$ in equation (2) we get

$x = 0 a n d x = 4$

Hence, critical point is $(0, 6) a n d (4, 6)$ .

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Find and classify the critical points of z = (x – 4x) (y 4.x Local maximums: DNE Local minimums: DNE Saddle points: DNE For each classification, enter a list of ordered pairs (x, y) where the r are no points for a classification, enter DNE,