Answer to Question #160262 in Differential Equations for Priya

Question

The function f(x,y,z)=max{y/x} is a homogeneous function on R².is it true or false.give reasons for your answer

Answer 1

$Solution:~ False. \\~~~~~ A ~homogeneous ~ real ~ valued~ function~ of ~two ~variable ~ x ~ and ~y ~ \\is ~a ~real ~ valued~ function~ that ~satisfies ~the ~condition~ f(rx,ry)=r^k f(x,y), \\ for ~some ~constant ~ k~ and ~all ~real ~ number ~ r. \\~~~Now,~ the ~given ~ real ~valued ~ function ~ of ~ two ~variable~ is ~ f(x,y)=max\{\frac{y}{x},x\} \\Putting ~ x=rx, ~ and ~ y=ry~ , we ~get, \\f(rx,ry)=max\{\frac{ry}{rx},rx\} \\~~~~~~~~~~~~~~~~~=max\{\frac{y}{x},rx\} \\~~~~~~~~~~~~~~~~~\neq r^k f(x,y)~ ,for ~some ~constant ~ k~ and ~all ~ real ~ number ~ r. \\Hence ,f~ is ~not ~ a ~ homogeneous ~function.$