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1. Homework Statement

[itex] \text{Let f be a differentiable function from ℝ^2 to ℝ that satisfies:}\ [/itex]

[itex]1) f(x,y)=0\ \text{for all}\ (x,y)\ \text{in the circumference}\ x^2+y^2=2[/itex]

[itex]2) \text{If we consider the function g from ℝ to ℝ^2 given by g(t)=(t+1,e^t), then the function fog has a relative minimum at t=0}\ [/itex]

[itex] \text{Find the gradient of f at the point (1,1)}\ [/itex]

The attempt at a solution

By the chain rule, I know that 0= D(fog(0))= D(f(g(0))g'(0)=([itex]\frac{\partial f}{\partial x}[/itex](1,1),[itex]\frac{\partial f}{\partial y}[/itex](1,1))[itex]

\begin{pmatrix}

1\\

1

\end{pmatrix}

[/itex]

[itex] \text{Doing matrix multiplication, I get the equation}\ [/itex][itex]\frac{\partial f}{\partial x}[/itex][itex](1,1)[/itex] + [itex]\frac{\partial f}{\partial y}[/itex][itex](1,1)=0[/itex]

So, I could use the information of 2) but I don't know how to use the data on 1). The only thing I know with 2) is that the partial derivatives satisfy [itex]\frac{\partial f}{\partial x}[/itex][itex](1,1)[/itex]=-[itex]\frac{\partial f}{\partial y}[/itex][itex](1,1)[/itex]. Could anyone tell me how to use the information given in 1)?

I'm trying to use latex but I'm clumsy, sorry about that, I'll try to improve my latex skills in the near future.

[itex] \text{Let f be a differentiable function from ℝ^2 to ℝ that satisfies:}\ [/itex]

[itex]1) f(x,y)=0\ \text{for all}\ (x,y)\ \text{in the circumference}\ x^2+y^2=2[/itex]

[itex]2) \text{If we consider the function g from ℝ to ℝ^2 given by g(t)=(t+1,e^t), then the function fog has a relative minimum at t=0}\ [/itex]

[itex] \text{Find the gradient of f at the point (1,1)}\ [/itex]

The attempt at a solution

By the chain rule, I know that 0= D(fog(0))= D(f(g(0))g'(0)=([itex]\frac{\partial f}{\partial x}[/itex](1,1),[itex]\frac{\partial f}{\partial y}[/itex](1,1))[itex]

\begin{pmatrix}

1\\

1

\end{pmatrix}

[/itex]

[itex] \text{Doing matrix multiplication, I get the equation}\ [/itex][itex]\frac{\partial f}{\partial x}[/itex][itex](1,1)[/itex] + [itex]\frac{\partial f}{\partial y}[/itex][itex](1,1)=0[/itex]

So, I could use the information of 2) but I don't know how to use the data on 1). The only thing I know with 2) is that the partial derivatives satisfy [itex]\frac{\partial f}{\partial x}[/itex][itex](1,1)[/itex]=-[itex]\frac{\partial f}{\partial y}[/itex][itex](1,1)[/itex]. Could anyone tell me how to use the information given in 1)?

I'm trying to use latex but I'm clumsy, sorry about that, I'll try to improve my latex skills in the near future.

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