Existence of directional derivative

songoku
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Homework Statement
Please see below
Relevant Equations
Partial derivative

Direction derivative in the direction of unit vector u = <a, b>:
Du f(x,y) = fx (x,y) a + fy (x,y) b
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My attempt:
I have proved (i), it is continuous since ##\lim_{(x,y)\rightarrow (0,0)}=f(0,0)##

I also have shown the partial derivative exists for (ii), where ##f_x=0## and ##f_y=0##

I have a problem with the directional derivative. Taking u = <a, b> , I got:
$$Du =\frac{\sqrt[3] y}{3 \sqrt[3] {x^2}}a+\frac{\sqrt[3] x}{3 \sqrt[3] {y^2}}b$$

Then how to check whether the directional derivative exists or not?

Thanks
 
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The directional derivative at ##[0,0]## in direction ##[a,b]## (has to be zero since both partial directions are and) is given by
$$
D_{[0,0]}(f)\cdot [a,b]=0=\lim_{h\to 0}\dfrac{f([0,0]+h[a,b])-f([0,0])}{h}
$$
What do you get on the right-hand side for ##f([x,y])= \sqrt[3]{xy}##?
 
fresh_42 said:
The directional derivative at ##[0,0]## in direction ##[a,b]## (has to be zero since both partial directions are and) is given by
$$
D_{[0,0]}(f)\cdot [a,b]=0=\lim_{h\to 0}\dfrac{f([0,0]+h[a,b])-f([0,0])}{h}
$$
What do you get on the right-hand side for ##f([x,y])= \sqrt[3]{xy}##?
$$0=\lim_{h\to 0}\frac{\sqrt[3]{(h^{2}ab)}-0}{h}$$
$$0=\lim_{h\to 0}\frac{(ab)^{\frac{1}{3}}}{h^{\frac{1}{3}}}$$

The limit can only exist if ##(ab)^{\frac{1}{3}}=0## so either ##a=0## or ##b=0## and since ##u## is unit vector, if ##a=0## then ##b=\pm 1## and vice versa

Is my working correct? Thanks
 
Yes, that is correct.
 
Thank you very much fresh_42
 
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