o g(x1,...,x,) = e"2 X% (Exponential function) (a) Find the gradient and the Hessian of the above functions at any point (z1, ..., z, ). rieie: % and any n for g. Then use the function gradient and hessian on both f and g when n =75 . (b) Using the same values of n as in (a), apply the first-order necessary condition to determine the candidates @ for local minima of the above functions. (c) Evaluate the Hessian matrices obtained in (a) at the points . Then verify the points T are local minima by using the second-order sufficient conditions. What properties did you check on the Hessian matrices to verify they are indeed positive definite matrices at the points 7 (d) Are the points T found above global minima? Justify your answer. Part a ) g(X1, X2, .., In) = -e-2Er, x? ax1 09 ( x1 , 2 , .., n) = ale 2(2]+..+x2) 09 ( 21, 2 2 , .., In) = x20 2(2]+..+22) (1 - 27)e -3(23+..+2, ) - *1ze 2 (20 ,+ .. + 2 2 ) (1 - 23 )e 2(23+..+2, ) . . . -Cline z (20 ] +.. +x2 ) 7 H, (x1 , .. n) = - 13e 2 (2] +..+2? ) . . (1 - x3)e -2(2]+..+x?) . . . -Clone z(2]+..+22) - zane 2 ( 2 , +.. + 2 2 ) -3age 2 (2] +..+x2 ) ... (1 -x2)e 2(23+..+22) . . - 15 e 2(2 7 +.. +23) (1 - x3)e 2(27+..+23) H, (21, ..25) = . . . - 2age 2 (2 ] +..+x3) -X2age 2(21+..+x3) . . . -3age 2 (2]+..+x3) - 2age 2(2 1 +..+x3 )