how to find critical points calc 3

A critical point of a multivariable part is a betoken where the partial derivatives of first guild of this function are equal to zero. Examples with detailed solution on how to notice the disquisitional points of a function with 2 variables are presented.
More Optimization Bug with Functions of Two Variables in this web site.

Examples with Detailed Solutions

Example one

Find the critical betoken(s) of part f defined past
f(ten , y) = xⁱⁱ + yⁱⁱ

Solution to Case one:
We beginning detect the offset guild partial derivatives.
f₁₀(x,y) = 2x
f_y(10,y) = 2y
We now solve the following equations f_x(x,y) = 0 and f_y(x,y) = 0 simultaneously.
f_x(x,y) = 2x = 0
f_y(x,y) = 2y = 0
The solution to the higher up organisation of equations is the ordered pair (0,0).
Below is the graph of f(x , y) = ten² + yⁱⁱ and information technology looks that at the disquisitional point (0,0) f has a minimum value.

Example 2

Find the critical point(s) of part f defined by
f(ten , y) = x² - y²

Solution to Case 2:
Find the starting time club fractional derivatives of role f.
f_x(x,y) = 2x
f_y(x,y) = -2y
Solve the following equations f_x(x,y) = 0 and f_y(x,y) = 0 simultaneously.
f₁₀(x,y) = 2x = 0
f_y(ten,y) = - 2y = 0
The solution is the ordered pair (0,0).
The graph of f(10 , y) = x^two - y² is shown below. f is curving down in the y direction and curving upwardly in the x management. f is stationary at the point (0,0) but there is no extremum (maximum or minimum). (0,0) is called a saddle bespeak considering there is neither a relative maximum nor a relative minimum and the surface shut to (0,0) looks like a saddle.

Example 3

Find the critical point(s) of function f defined past
f(10 , y) = - ten² - yⁱⁱ

Solution to Example iii:
We first discover the first guild partial derivatives.
f_x(ten,y) = - 2x
f_y(ten,y) = - 2y
We now solve the following equations f_x(10,y) = 0 and f_y(x,y) = 0 simultaneously.
f_x(x,y) = - 2x = 0
f_y(ten,y) = - 2y = 0
The solution to the higher up system of equations is the ordered pair (0,0).
The graph of f(x , y) = - x² - y² is shown beneath and information technology has a relative maximum.

Example 4

Find the disquisitional indicate(due south) of divers by
f(x , y) = x³ + 3x^two - 9x + y^three -12y

Solution to Example 4:
The first guild partial derivatives are given by
f₁₀(10,y) = 3x² + 6x - ix
f_y(x,y) = 3y² - 12
We now solve the equations f_x(x,y) = 0 and f_y(10,y) = 0 simultaneously.
3x² + 6x - nine = 0
3yⁱⁱ - 12 = 0
The solutions, which are the critical points, to the above system of equations are given by
(1,ii) , (1,-ii) , (-3,2) , (-iii,-2)

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Exercises

Find, if any, the disquisitional points to the functions below.
1. f(x , y) = 3xy - ten ⁱⁱⁱ - y ⁱⁱⁱ
2. f(x , y) = e ^{(- x2 - y2 + 2x - 2y - 2)}
3. f(x , y) = (i/2)x ² + y ⁱⁱⁱ - 3xy - 4x + 2
4. f(x , y) = 10 ³ + y ^three + 2x + 6y

Answers to the Higher up Exercises

ane. (0,0) , (1,ane)
2. (1,-1)
3. (16,4) , (1,-ane)
4. no critical points.

More on Partial Derivatives and Multivariable Functions

Multivariable Functions

Source: https://www.analyzemath.com/calculus/multivariable/critical_points.html

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