3D System — Extreme Points z=f(x,y)

Finding Extreme Points By Differentiation— #PySeries#Episode 24

Hi, in this post you’ll learn how to use a process known as differentiation to find a function’s derivative. In addition, you’ll learn other important calculus concepts known as critical points, extreme values. And More, we will Plotting a NumPy Polynomial with Matplotlib.

We are going to use Python and Geogebra.

Welcome!

01PyEx — Given the function z to respect to x and y, determine the extreme points:

Fig 1. The point that we will calculate is A; C is the projection showing the Minimum in 3D; Please, keep reading:); See vid & geo!

Open a new notebook in your Colab now!

Let’s get down to work! Take the Derivatives of z to the respect of x and them y

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import sympy as sy
x, y = sy.symbols('x y')
Z = 3*x**2 + 2*x*y + y**2 + 10*x + 2*y + 1
Zx = sy.diff(Z, x)
Zy = sy.diff(Z, y)
print(f'Zx = {Zx}\nZy = {Zy}')
Zx = 6*x + 2*y + 10 
Zy = 2*x + 2*y + 2

The Equation again!

The first differentiation gives us 2 equation: a Linear System. Let’s solve it:

A = np.array([[6, 2], [2, 2]])
b = np.array([-10, -2])
x = np.linalg.solve(A,b)
print(x)
[-2.  1.]

So the points are x=−2; y=1

Here are The 3 Equations (y1, y2 and y3):

x, y = sy.symbols('x y')
y1 = 3*x**2 + 2*x*y + y**2 + 10*x + 2*y + 1
y2 = 6*x + 2*y + 10
y3 = 2*x + 2*y + 2

Calculation of The First and the Second Partial Differentiation:

Discriminant — Formula

D=Zxx ∗ Zyy −( Zxy)2

It involves the second pure partial derivatives in product, subtracting the second mixed derivative squared

Zxx

#differentiate y1 with respect to x
Zx = sy.diff(y1, x)
#returns resulting expression in factored form
#this matters to our specific problem and looking
#for some kind of structure in the expression
sy.factor(Zx)
2(3x+y+5)

Now the second partial:

Zxx = sy.diff(Zx, x)
print(f'Zxx={sy.factor(Zxx)}')
Zxx=6

Zyy

#differentiate Zy with respect to y
Zy = sy.diff(y1, y)
#returns resulting expression in factored form
#this matters to our specific problem and looking
#for some kind of structure in the expression
sy.factor(Zy)
2(x+y+1)

And the second:

Zyy = sy.diff(Zy, y)
print(f'Zyy={sy.factor(Zyy)}')
Zyy=2

Zxy

#differentiate y1 with respect to x
Zx = sy.diff(y1, x)
#returns resulting expression in factored form
#this matters to our specific problem and looking
#for some kind of structure in the expression
sy.factor(Zx)
2(3x+y+5)

The second:

Zxy = sy.diff(Zx, y)
print(f'Zxy={sy.factor(Zxy)}')
Zxy=2

Discriminant — Formula

D= Zxx ∗ Zyy − (Zxy)^2
D= 6 ∗ 2 − (2)^2 = 8
D = 8

As D>0 and Zxx>0

Interpretation:

If D > 0 and Zxx < 0 ↦ MAX

If D > 0 and Zxx > 0 ↦ MIN

If D < 0 ↦ SADDLE

If D = 0 ↦NOT−CONCLUSIVE

So the points (-2,1) represents the Minimum

(see the graph above)

Plotting a polynomial in Geogebra

Equations:

z = 3x2+2xy+2y+10x+2y+1
Zx = 6x+2y+10=0 ↦ 6x+2y=−10
Zy = 2x+2y+2=0 ↦ 2x+2y=−2

See these Graph in Geogebra and video in Youtube:

Please, Watch the video to understand the three-dimensional figure and the point location.

You notice that the point found was on the xy plane. If you project the point (A) along the z-axis, you will see that the location of the point fits perfectly in 3D figure (C).

Fig 1. The point that we will calculate is A (-2,1); C (-2,1,-6.4) is the projection showed the Minimum. Vid here Geo there:)!

print("There you have it! Thank you very much! Python is awesome, doesn't it?")
There you have it! Thank you very much! Python is awesome, doesn't it?

That’s All for this lecture!

See you in the next Python Episode!

See answer below!

Bye!!!!

Colab File link:)

Google Drive link:)

Geogebra Solution link:)

Video link:)

Credits & References

Geogebra Solution by J3

Youtube vid by J3

Decimal to Fraction Calculator by calculatorsoup.com

Exercícios Resolvidos Assunto: Integral Dupla (pdf) by Universidade Federal Fluminense (Brazil — Niterói — RJ)

Welcome to Calculus with Python’s documentation! — Calculus with Python by calc-again.readthedocs.io

LaTeX/Mathematics by wikibooks.org

Eddie Woo Vids by Eddie Woo

Taking Derivatives in Python by Dario Radečić

Plotting a polynomial in Python by stackoverflow.com

Three-Dimensional Plotting in Matplotlib by jakevdp.github.io

Finding Extreme Points by https://www.dataquest.io/m/159-finding-extreme-points/

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