Please can you help with this. I need a detailed explaination

Question

Use Lagrange multipliers to find the extreme values of f(x, y, z) = yz + zx subject to the constraints yz = 1 and

x^2 + z^2 = 1.

Answer 1

Let ∇f = λ∇g +μ∇h where g= yz=1 and h=x² +z² =1. Hence taking partial derivatives with respect to x,y,z gives

z= 0λ +2μx and z=λz +0μ and y+x= λy +2μz. This set of equations simplifies to λz=z giving λ=1 and x= 2μz and z=2μx so that x=z. So x=2μ(2μx) or 1=4μ² and μ =±1/2. Using the form of h and substituting for x gives 2z²=1 or z=±1/2 =x and using the form of g gives y=1/z = ±√2.. The possibilities for extremum are (1/√2,√2,1/√2), (1/√2,-√2, 1/√2), ( -1/√2, -√2,-1/√2) and (-1/√2,√2,-1/√2). Evaluating f at each of these sets of values gives 0 and 3/2 as the extremum values.