Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
a- if f is a function of x and y and a is a real number, then f(ax, ay)= af(x,y).
b- if fx(a,b) < 0, then f is decreasing with respect to x near (a,b).

Answer 1

(a) is certainly not true in general. While it is true for linear functions, it is otherwise not true.

f(x,y) = x^2/(y^2+1)
f(ax,ay) = (a^2x^2)/(a^y^2+1)

or

f(x,y) = sin(x+y)
f(ax,ay) = sin a(x+y)

or

f(x,y) = e^x * ln(y)
f(ax,ay) = a^ax * ln(ay) = e^a * e^x * (lna + lny)


(b) I assume fx means the partial with respects to x. That's true. The derivative is the slope of the curve in the intersection of f(x,y) and the plane y=b.