If f (x, y) is a homogeneous function of degree n in x and y and hascontinuous first and second order partial derivatives then(a) x20?a²f+ 2xy+ y2 °дхдуду?= n(n – 1)f(x,y)ax2(b)x2 02əx2a?fa2f= nf(x,y)ду?+ 2xy+дхду(c)x2 32a²f+ 2xy(n – 1)f(x,y)+ y2Əx2|дхдуду?(d)x + 2xy+ y = f(x,y)|a2fƏx2дхдуду2

This question is based on Euler's theorem statement.

Answered: If f (x, y) is a homogeneous function…

If f (x, y) is a homogeneous function of degree n in x and y and has continuous first and second order partial derivatives then

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.2: Partial Derivatives

Problem 28E

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Question

If f (x, y) is a homogeneous function of degree n in x and y and has
continuous first and second order partial derivatives then
(a) x20?
a²f
+ 2xy
+ y2 °
дхду
ду?
= n(n – 1)f(x,y)
ax2
(b)x2 02
əx2
a?f
a2f
= nf(x,y)
ду?
+ 2xy
+
дхду
(c)x2 32
a²f
+ 2xy
(n – 1)f(x,y)
+ y2
Əx2
|
дхду
ду?
(d)x + 2xy+ y = f(x,y)|
a2f
Əx2
дхду
ду2

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