Multivariable chain rule and differentiability

[tomelwood]

Homework Statement

Hi I'm currently trying to revise for a Calculus exam, and have very little idea of how to do the following:
Let f be defined by f(x,y) = (y+e^x, sin(x+y))
Let g be of class C2 (twice differentiable with continuous second derivatives) with grad(g)(1,0) = (1,-1) and Hg(1,0) = $$\left(\stackrel{2}{0}\stackrel{0}{0}\right)$$
Consider F=gof (g composition f)
Prove that F is differentiable on the whole of $$\textbf{R}$$$$^{2}$$. Calculate dF(x,y) with respect to the partial derivatives of g.
Study if F has an extreme at (0,0)

Homework Equations

The Attempt at a Solution

Like I said, I'm not really sure what to do here. I know that that Hessian matrix means that g$$_{xx}$$ (1,0) = 2, and the other second derivatives are 0, and that the grad being (1,-1) means that g$$_{x}$$(1,0) = 1 and g$$_{y}$$(1,0) = -1, but I don't know what I can do with this.

Or is it just a case of saying that f is differentiable on the whole plane (since y+e^x and sin(x+y) are both differentiable on the whole plane, aren't they?), and g is of class C2, so also differentiable on the whole plane??
But then how do I continue to calculate the derivative of F? If the above even makes sense, which I doubt.

Any help would be greatly appreciated!

(Also, while I'm here, is the tangent plane to f(x,y) = xy at (0,0) the plane z=0? Thanks.)

 

[mighty2000]

Thank you for your question and for sharing your thoughts on the problem. I can understand your confusion, as this problem does involve some advanced concepts in multivariable calculus. Let me try to break it down for you and provide some guidance on how to approach it.

Firstly, as you mentioned, both f(x,y) and g(x,y) are differentiable on the whole plane, so we can assume that F(x,y) will also be differentiable on the whole plane. However, we need to show this rigorously by using the definition of differentiability.

To prove that F is differentiable on the whole plane, we need to show that the limit of (F(x,y)-F(a,b))/(x-a) and (F(x,y)-F(a,b))/(y-b) exists as (x,y) approaches (a,b). This limit will be equal to the partial derivatives of F with respect to x and y, respectively.

To calculate these partial derivatives, we can use the chain rule. For example, to find the partial derivative of F with respect to x, we can write:

∂F/∂x = ∂g/∂x * ∂f/∂x + ∂g/∂y * ∂f/∂y

We already know the values of ∂g/∂x and ∂g/∂y at (1,0) from the given information. To find ∂f/∂x and ∂f/∂y, we can use the partial derivatives of f(x,y) that are given in the problem. Similarly, we can find ∂F/∂y using the same approach.

Once we have found these partial derivatives, we can use them to construct the Jacobian matrix of F, which will be a 2x2 matrix. This matrix will represent the linear transformation given by the partial derivatives of F at any point (a,b). We can then use this matrix to find the directional derivative of F in any direction at any point.

Now, to study if F has an extreme at (0,0), we can use the second derivative test. We already know the value of the Hessian matrix of g at (1,0) from the given information. We can use this to find the Hessian matrix of F at (0,0) by using the chain rule again. Once we have the H