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**1. Homework Statement**

Let ##g(x,y)=\sqrt[3]{xy}##

a) Is ##g## continuous at ##(0,0)##?

b) Calculate ##\frac {\partial g}{\partial x}## and ##\frac{\partial g}{\partial y}## when ##xy \neq 0##

c) Show that ##g_{x}(0,0)## and ##g_{y}(0,0)## exist by supplying values for them.

d) Are ##\frac {\partial g}{\partial x}## and ##\frac {\partial g}{\partial y}## continuous at ##(0,0)##?

e) Does the graph of ##z=g(x,y)## have a tangent plane at ##(0,0)##? You might consider creating a graph of this surface.

f) Is ##g## differentiable at ##(0,0)##?

**2. Homework Equations**

**3. The Attempt at a Solution**

a) $$lim_{y \rightarrow 0 \space along\space x=0}\space g(x,y) =0$$

$$lim_{x \rightarrow 0 \space along\space y=0}\space g(x,y)=0$$

$$lim_{y=mx\rightarrow 0} = 0$$

Therefore limit exists and the limit is ##0##. Since $$g(0,0)=0$$ and is equal to limit, ##g## is continuous at ##(0,0)##.

b) $$\frac {\partial g}{\partial x}=\frac{\frac{1}{3}y^\frac{1}{3}}{x^\frac{2}{3}}$$

$$\frac{\partial g}{\partial y}=\frac{\frac{1}{3}x^\frac{1}{3}}{y^\frac{2}{3}}$$

c) I'm not sure what they meant by supplying values for them. But, what I did was giving the value of zero to both of the partials. I'm not sure how to do this part of the question.

d) For $$\frac {\partial g}{\partial x}$$, $$lim_{x \rightarrow 0\space along\space y=0} \frac{\partial g}{\partial x}$$

$$= \frac{0}{x^\frac{2}{3}} = 0$$

$$lim_{y \rightarrow 0\space along\space x=0} \frac{\partial g}{\partial x}$$ is undefined.

Therefore, ##g_{x}## is not continuous at ##(0,0)##. The same result is obtained for ##g_{y}##.

e) I don't know. Although in (b), it is shown that ##g_{x}## and ##g_{y}## exist at ##(0,0)##, since the partials are not continuous, I would say that there is no tangent plane at the origin.

f) Yes, I would say the function is differentiable at origin, because I can differentiate it using the chain rule. What has differentiability got to do with continuity? Why is this question asked? Does differentiability imply continuity?

It is a lot of questions, but I hope someone can help me out here.

Thanks.

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