How to determine where a multivariable function isn't differentiable?

Question

consider $f(x,y) = \max(x+2y, x^2+y^2)$, how do i find the points at which this isn't differentiable.

I've tried taking the partial derivative of $x$ and $y$, for general points, but i get stuck with an equation that i can't seem to simplify, is this the wrong method?

Answer 1

Hint.

The attached plot shows $f(x,y)$

Considering the change of coordinates

$$ \{x,y,z\}\Leftrightarrow\{r \sin (\phi ) \cos (\theta ),r \sin (\phi ) \sin (\theta ),r \cos (\phi )\} $$

in the equations

$$ x^2+y^2 = z\\ x + 2y = z $$

we get the parameterization for the intersection curve as

$$ \left\{\cos (\theta ) (2 \sin (\theta )+\cos (\theta )),\sin (\theta ) (2 \sin (\theta )+\cos (\theta )),(2 \sin (\theta )+\cos (\theta ))^2\right\} $$

for $0 \le \theta < 2\pi$ shown in blue.

Answer 2

To get the function differentiable at the intersection of both surfaces, you need that the derivatives are the same, $(1,2)=(2x,2y)$. This is only the case for the point $(x,y)=(\frac12,1)$. Now $x+2y=\frac94$ and $x^2+y^2=\frac54$, so that this is inside the intersection circle where the first function is dominant, not on its boundary.