Non-differentiability of a function of two variables at a point

Question

I have problems understanding this:

Function $\;g(x,y)\;$ is given, for which

a) $\;g_x(0,0)=7\;$

b) $\;g(t+2t^2,\sin3t+4t^2)=5e^t\;$

c) $\;\lim_{t\to 0}\frac{g(t,2t)-g(3t,4t)}t=10\;$

They ask to show there's a point for which $\;g\;$ isn't differentiable and what is that point.

I use b) to have that $g(t+2t^2,\sin3t+4t^2)=5e^t\to 5\;$ with $\;t\to 0\;$, and then it can't be continuous at origin since $\;g(0,0)=7\;$ and then not differentiable, but then I don't get what condition c) tells.

Answer 1

Since two of three conditions have to do with $(0,0)$, the only reasonable conjecture for the point of non-differentiability is $(0,0)$.

So, suppose $g$ is differentiable at $(0,0)$, meaning there are constants $a,b$ such that $$ \frac{g(x,y)-g(0,0)-ax-by}{\sqrt{x^2+y^2}} \to 0 \quad \text{as } (x,y)\to (0,0) \tag{1} $$

Plugging (a),(b),(c) into (1), you should be able to obtain that

• $a=7$
• $a+3b=5$
• $-2a -2b = 10$

Which is probably not consistent...

If you know the multivariable chain rule, use that instead of (1).