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{\bf Math 273}: {\bf Homework \#1, due on Friday, October 16} 

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{\bf 1.} Compute the gradient $\nabla f(x)$ and Hessian $\nabla^2 f(x)$ of the function 
$$f(x)=100(x_2-x_1^2)^2+(1-x_1)^2.$$ 
Show that $x^*=(1,1)^T$ is the only local minimizer of this function, and that the Hessian matrix at that point is positive definite. 

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{\bf 2.} Let $a$ be a given $n$-vector, and $A$ be a given $n\times n$ symmetric matrix. Compute the gradient and Hessian of $f_1(x)=a^Tx$ and $f_2(x)=x^TAx$. 

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{\bf 3.} Suppose that $f$ is a convex function. Show that the set of global minimizers of $f$ is a convex set. 

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{\bf 4.} Suppose that $\hat f(z)=f(x)$, where $x=Sz+s$ for some $S\in R^{n\times n}$ and $s\in R^n$. Show that 
$$\nabla \hat f(z)=S^T\nabla f(x),\ \ \ \ \nabla ^2\hat f(z)=S^T\nabla^2 f(x)S.$$ 

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{\bf 5.} (Directional derivative) Let $f:R^n\rightarrow R$ be continuously differentiable. Show that 
$$\lim_{\epsilon\rightarrow0}\frac{f(x+\epsilon p)-f(x)}{\epsilon}=\nabla f(x)^Tp.$$

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{\bf 6.} Consider the minimization problem 
$$\inf_{u}\int_{\Omega}L(x,y,u,u_x,u_y)dxdy,\ \ \ u=g\mbox{ on }\partial\Omega,$$
where $g$ is given. Let $\Omega$ be a bounded region in $R^2$, with $\partial\Omega$ its boundary. Show that a sufficiently smooth solution $u$ satisfies the Euler-Lagrange equation 
$$\frac{\partial}{\partial x} L_{u_x}(P)+\frac{\partial}{\partial y}L_{u_y}(P)-L_u(P)=0$$
on $\Omega$, where $P=(x,y,u(x,y),u_x(x,y),u_y(x,y))$. 

Apply the above result to the case when $L(x,y,u_x,u_y)=u_x^2+u_y^2-2fu$. 

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