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Which one of the following function(s) is/are homogeneous?A. `f(x,y) = (x-y)/(x^(2)+y^(2))`B. `f(x,y) = x^(1/3)y^(-2/3) tan^(-1)x/y`C. `f(x,y) = x(" ln "sqrtx^(2)+y^(2))-" ln "y+ye^(x//y)`D. `f(x,y) = x[" ln "(2x^(2)+y^(2))x -" ln "(x+y)]+y^(2)tan(x+2y)/(3x-y)` |
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Answer» Correct Answer - A::B::C a) `f(lambdax,lambday)=(lambda(x-y))/(lambda^(2)(x^(2)+y^(2)))=lambda^(-1)f(x,y)` Thus, it is homogenous of degree `-1` b) `f(lambdax,lambday)=(lambdax)^(1//3)(lambday)^(-2//3)tan^(-1)x/y` `=lambda^(-1//3)x^(1//3)tan^(-1)x/y` `=lambda^(-1//3)f(x,y)` c) `f(lambdax,lambday) = lambdax("ln"sqrt(lambda^(2)(x^(2)+y^(2))-"ln "lambday))+lambdaye^(x//y)` `=lambdax["ln"((lambdasqrt(x^(2)+y^(2)))/(lambday))]+lambdaye^(x//y)` `lambda[x("ln "sqrt(x^(2)+y^(2))-"ln"y)+ye^(x//y)]` `=lambdaf(x,y)` Thus, it is homogeneous. d) `f(lambdax,lambday)=lambdax["ln "(2lambda^(2)x^(2)+lambda^(2)y^(2))(lambdaxlambda(x+y))]+lambda^(2)x^(2)tan(x+2y)/(3x-y)` `=lambda x["ln "(2x^(2)+y^(2))/(x(x+y))]+lambda^(2)x^(2)tan(x+2y)/(3x-y)` Thus, it is non-homogeneous. |
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