I'm having an exam of multivariate calculus . I'm not able to solve a question.
The Question is:
$$f(x,y) = e^x (\cos y +x \sin y)$$ Show that $F_{xy} = F_{yx}$ at Origin by Definition
Hint : use L-hospital use to solve the question.
Here is the link of the snap of my register copy that explains what I've done. https://i.stack.imgur.com/AjZYc.jpg
As in the snap, I'm stuck as if I apply the limit, that makes the denominator zero and infinity should not be a correct ans I believe.
I'm stuck as if I apply the limit, that makes the denominator zero and infinity should not be a correct ans I believe.
I would really appreciate if someone would help me understand how to solve this problem.
Thanks
Expanding first: $$f(x,y) = e^x \cos y + x e^x \sin y$$ $$f_{xy} = (f_x)_y= \frac {{\partial }^2f}{{\partial x}{\partial y}}$$ $$f_{yx} = (f_y)_x= \frac {{\partial }^2f}{{\partial y}{\partial x}}$$ $$f_{x} = e^x \cos y + e^x \sin y +x e^x \sin y$$ $$f_{xy} = -e^x \sin y + e^x\cos y + x e^x \cos y$$ Evaluating this at $(0,0)$ gives: $f_{xy}= 1$ Now : $$f_{y} = -e^x \sin y +xe^x \cos y$$ $$f_{yx}= - e^x \sin y + e^x \cos y + xe^x \cos y$$ Evaluating this at $(0,0)$ gives: $f_{yx} =1$. Hence: $f_{xy}=f_{yx}$ at the origin.