- #1
tomelwood
- 34
- 0
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) = [tex]\left(\stackrel{2}{0}\stackrel{0}{0}\right)[/tex]
Consider F=gof (g composition f)
Prove that F is differentiable on the whole of [tex]\textbf{R}[/tex][tex]^{2}[/tex]. 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[tex]_{xx}[/tex] (1,0) = 2, and the other second derivatives are 0, and that the grad being (1,-1) means that g[tex]_{x}[/tex](1,0) = 1 and g[tex]_{y}[/tex](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.)