• Calculate the gradient of the function f(x1,x2,x3) = 2x51x32x43 +x1x2 +x2x3 at (1,1,1).
• Let f(x1,x2) = (x1 +x2)4 ?8(x1 +x2)2. Find all the local minimas and local maximas

  of this function. Guess what the graph of this function looks like.

• Let x ? R2. Consider the function f(x) = xTAx. If A is an invertible matrix, then prove that this function has only one stationary point at 0(stationary points are the points at which the gradient is zero). Give an A for which 0 is the minimizer of f(x). Give an A for which 0 is neither a maximizer nor a minimizer of f(x).
• Complete the proof of Lemma 7 in Lecture note 2 (posted on our course website).
• Let x ? Rp and find the gradient of the following functions:

  (a) f1(x) = (xT Ax)2, where A is an n × n matrix. (b) f2(x) = (xT Ax)n, where A is an n × n matrix.

• Let&nbsp;A&nbsp;?&nbsp;Rn×p&nbsp;denote a fat matrix, i.e.,&nbsp;n < p. Explain why we should expect the equation&nbsp;y&nbsp;=&nbsp;Ax&nbsp;to have infinitely many solutions. Among all those solutions we would like to find the one with minimum Euclidean norm, i.e., we want to find the solution with the smallest&nbsp;xT&nbsp;x. Find that solution and prove your answer.
• Plotthefunctionf(x1,x2)=(x1?1)2+(x2?1)2+x1x2&nbsp;inR.Assumethatx1&nbsp;?[?2,2] and&nbsp;x2&nbsp;?&nbsp;[?2,&nbsp;2]. Before doing this problem you may want to study the commands “outer” and “persp” in R.

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