Question

Need help with this two questions

1. Consider the isoperimetric problem: = / yV1+y2da= min, y(0) y(a)0 subject to the constraint J = Jo In class we showed that this problem could be written as the solution of the ODE A VI2 where A is a constant and A is the Lagrange multiplier a) Show very clearly that the general solution of this equation can be written as B y A - Acosh b) Determine the values of A, A, and B. 2. Using both the "direct" and Lagrange multiplier approaches, minimize the function f(r, y)22 subject to g(x, y) 2x y - 2 = 0. Evaluate the Hessian matrix to test whether or not the stationary point corresponds to a minimum global

Answer 1

b) y = x-A cash (x-1 . Ylo) = 0 = x= A cosh B 2-0 y (a) = 0 = x= A cash aß - 6 Companing o und & we get 16 Ja= 2B 1 TB 2 If id x= Alosh a ! it sin? n(m. due s Cash (n-3) da- e. | 22A ( 1 ) ( -3. Sin B ) : 4 = A ( 2 sinh a lze - 15 A los hátt * A can be obtained bysaving this nom lineer equation

(1) L. 7 (yrity sex Vity) da FU,72 Virgiz dr = 2 Beltrami 1o) - glas, J. It is independent of x, we can ux identit - Y' IF A ay @ naging 1001+ y = fer- a À Jy - 1 1 - 4 Sazo7+c) B constat Comtat Yux ñ losh ne y = x - A cash (122)

2) Directi- Slx,y)= x+y" g(x,y) - anty-2=0 4- 2-22 f(x)= x² + 4 (221) df - 22+ 8(2-1) = 0 7 101-5-0 -15 Hessian = [ser day I I ser tag i loz i - 2 242 is a hositive definite matrix. So, it is a miniman Lagrange The x 79 . - 1 21,2y) = x(2, 1) ar = 2x 4y= 2x 20= 44 = 24 2a + 2 / 2 = 2 = 51 = 2