Show your work and be neat. Here is a Lagrange multiplier problem in R2. The obj

Question

Show your work and be neat. Here is a Lagrange multiplier problem in R2. The objective function is o(x) = x2-(x)2. The constraint is the unit circle, given by g(x)-0 for g (z) = 1x12-1. Find the max and min values of (x) subject to the constraint g(z) = O. The gradient of g you should know. Calculate the gradient of by linearization. Yan shoold get 3 points tLgr rtquation 2. Let u = (3,-5, 1) and set (x) = 1x12 + u·x. Let S be the surface defined by the level set {x : (r) 70) (a) Verify that u is on the surface. (b) Calculate the normal to S at u using the methods from class. (c) What is the equation of the tangent plane for S at u?

Explanation / Answer

u = (3,-5,1) and pi(x)= |x|^2 + u.x

S={x: pi(x) =70}

To verify that, u is on the surface. Find pi(u)

pi(u) =  |u|^2 + u .u = (sqrt(3*3+(-5)(-5)+1*1))^2 + (3*3+(-5)(-5)+1*1) = (sqrt(9+25+1))^2 + (9+25+1) =70

from the definition of S, u is on the surface S.

(b) Information not clear as method not mentioned clearly.

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