Answer to Question #119961 in Differential Equations for SUSANTO SARKAR

Question

The function f(x,y) = max {y/x,x} is a homogeneous function on R^2

Answer 1

Recall a fact : A function F(x,y) is said to be homogeneous function of degree n if

F(ax,ay) = aⁿ F(x,y)

Now in our problem F(x,y)= max{y/x,x}

It means F(x,y)=y/x OR F(x,y)=x

Case i) If F(x,y) =y/x

Now F(ax,ay)=ay/ax=y/x

=a⁰F(x,y)

In this case F(x,y) is a homogenous function of degree zero.

Case ii) If F(x,y)=x

Now F(ax,ay)=ax

=a¹F(x,y)

In this case F(x,y) is a homogeneous function of degree 1

Therefore by i) and ii) case we can say F(x,y) is a homogeneous function.