Question

If f is homogeneous of degree n, showthat
fx(tx,ty)= tn-1 fx(x,y)

Answer 1

To show that f is homogeneous of degree n, we can use the definition of homogeneity and the properties of the function f.

Step-by-step explanation:

To show that f(x(tx,ty)) = t^{n-1}f(x,y) for a function f that is homogeneous of degree n, we can use the definition of homogeneity and the properties of the function f as follows:

1. Start with the left-hand side of the equation: f(x(tx,ty))
2. Since f is homogeneous of degree n, we can apply the property f(kx,ky) = k^nf(x,y) for any constant k.
3. Using the property from step 2, we have: f(x(tx,ty)) = (tx)^nf(x,y) = t^nx^nf(x,y)
4. Finally, we can express t^n as t^{n-1}t and substitute it into the equation to get: f(x(tx,ty)) = t^{n-1}tx^nf(x,y) = t^{n-1}f(x,y)

Therefore, we have shown that f(x(tx,ty)) = t^{n-1}f(x,y) for a function f that is homogeneous of degree n.