Functions of Several Variables — Question 3

Question 3 – Solution

Strategy Substitute scaled inputs, then solve the level equation explicitly for $y$.

Step 1: Scaling Direct substitution gives $$F(sx,sy)=\frac{s^2xy}{s(x+y)}=sF(x,y).$$ Thus the output has degree-one scaling.

Step 2: Level curve Set $xy/(x+y)=c$: $$xy=cx+cy\Longrightarrow y(x-c)=cx\Longrightarrow\boxed{y=\frac{cx}{x-c}}.$$ For positive $x,y$, the denominator must be positive, so $x>c$; symmetry also implies $y>c$.

Step 3: Geometry Rewrite $y=c+c^2/(x-c)$. The asymptotes are $\boxed{x=c}$ and $\boxed{y=c}$, and the positive-input level is the branch with $x>c,y>c$. Substitution back into $F$ returns $c$.