Consider the following maximization problem: The unique stationary point is given by X* = maxx,y[tx + ky - x²/2 - y²/2] y* = The 1st leading principal minor of the Hessian evaluated at (x*,y*) is |H₁| = The 2nd leading principal minor of the Hessian evaluated at (x*,y*) is |H₂| =

Two waves on a string are given by the following functions: yı (x, 1) = 4 cos(201 – 30x) (cm) y2(x, 1) = -4 cos(201 + 30x) (cm) where x is in centimeters. The waves are said to interfere constructively when their superposition lys| = ly1 + y2|maximum, and they interfere destructively when lys| is a minimum. (a) What are the directions of propagation of waves y1 (x.f) and y2(x, t1)? (b) At t = (= (T/50) s, at what x location y do the two waves interfere constructively, and what is the corresponding value of ys|? (c) At t = (7/50) s, at what locationx do the two waves interfere destructively, and what is the corresponding value of lys?

from ( 0,1) to (2,5)**

3. Show that the functionsy₁ =e¯*, y₂ = xe¯*form a fundamental set for the DE y" +2y'+y=0. Show all work.