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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Using First Partial Derivatives In Exercises 41-44, find all values of x and y such that f x ( x , y ) = 0 and f y ( x , y ) = 0

simultaneously.

f ( x , y ) = 3 x 3 12 x y + y 3

To determine

To calculate: The all value of x and y from the function f(x,y)=3x312xy+y3 such that fx(x,y)=0 and fy(x,y)=0 simultaneously.

Explanation

Given information:

The provided function is f(x,y)=3x312xy+y3 and the condition is fx(x,y)=0 and fy(x,y)=0 simultaneously.

Formula used:

Consider the function z=f(x,y) then for the value of zx consider y to be constant and differentiate with respect to x and the value of zy consider x to be constant and differentiate with respect to y.

Calculation:

Consider the provided function is,

f(x,y)=3x312xy+y3

Partially derivative of the function f(x,y)=3x312xy+y3 with respect to x.

fx(x,y)=x(3x312xy+y3)=3x(x3)12yx(x)+y3x(1)=9x212y

Substitute 0 for fx(x,y).

9x212y=012y=9x2y=34x2 …...…... (1)

Partially derivative of the function f(x,y)=3x312xy+y3 with respect to y.

fy(x,y)=y(3x312xy+y3)=3x3y(1)12xy(y)+y(y3)=12x+3y2

Substitute 0 for fy(x,y)

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