Please help me with the following 3 questions:

MULTIVARIABLE CALCULUS AND UNCONSTRAINED OPTIMIZATION Problem 4. Let f : R" }> IR be a Cl function. Suppose that the directional derivatives of f at x* E R\" vanish along 71 linearly independent vectors v1, . . . , v\" E R": 1,108) = ~ -- = fined\") = 0- Show that X\"' is a stationary point of f. Problem 5. For each part, write down an explicit example of a function with the stated properties, or explain why no such function exists. a) C2 function f : R2 } R with a stationary point at (1,1) and 2 _ Hf(*l,*1) = [_4 0] b) C2 function f : R2 > R with a saddle point at (1,1) and HIH, 71) = [3 3] 1 [2 FALL '17 ANALYSIS AND OPTIMIZATION: HOMEWORK 2 c) C2 function 1:lR2 A R with a local min at (71, 71) and 2 0 Hf(71171) = [0 0] d) C2 function 1:lR2 > R with a local max at (1, 1) and 2 4 Hf('1:'1)=|:_4 0] e) C2 function f : R2 } R with 2 2 4 Problem 6. Consider the function f($syaz) : $2 + $21; + y2z + 2:2 42:. Find all stationary points. Use the second derivative test to determine if each one is a local ruin, local max, or saddle point (or state that the test cannot determine the answer)