1 Calculate the gradient of the function f(x1 x)-2X51X32X43 +X1X2 +X2X3 at (1, 1

Question

1 Calculate the gradient of the function f(x1 x)-2X51X32X43 +X1X2 +X2X3 at (1, 1,1). 2. Let f(x1-x2) = (x1 +x2)4-8(x1 +x2)2. Find all the local minimas and local maxim as of this function. Guess what the graph of this function looks like. 3. Letx E R2. Consider the function fx)-xTAx. IfA 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 i zero). Give an A for which 0 is the minimizer of fx). Give an A for which 0 is neither a maximizer nor a minimizer of fx). 4 Complete the proof of Lemma 7 in Lecture note 2 (posted on our course website) 5 Let x Rp and find the gradient of the following functions: (a) fi (x) = (xT Ax), where A is an n x n matrix. (b) f2(x) = (xT Ax. where A is an n × n matrix. 6. Let A Rnip denote a fat matrix i.e., n

Explanation / Answer

1)given f(x1,x2,x3)=2x15x23x34+x1x2+x2x3

gradient f=<2*5x14x23x34+1*x2+0,2*3x15x22x34+x1*1+1*x3,2*4x15x23x33+0+x2*1>

gradient f=<10x14x23x34+x2,6x15x22x34+x1+x3, 8x15x23x33+x2>

at (1,1,1)

gradient f=<10*141314 +1,6*151214+1+1, 8*151313+1>

gradient f=<10 +1,6 +1+1, 8 +1>

gradient f=<11,8,9>

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