Consider the problem of minimization of the following functi

Consider the problem of minimization of the following function for all x epsilon R^2. Integral(x)=(x_1-3x^2_2)(x_1-x^2_2) Determine all stationary points of integral. Determine which, if any, stationary points are local minima? Local maxima? Explain.

Solution

Let us write x₁= x and x₂ = y for the sake of convenience in typing.Then f ( x,y) = ( x -3y²)( x – y²) =x² – 4xy²+ 3y⁴ . Then, we have,

f/x = 2x – 4y² ; f/y = -8xy + 12y³ ; ² f/x² = 2 ; ² f/y² = -8x + 36y² ; ² f/xy = - 8y ; ² f/yx = -8y

We may observe that ² f/xy = ² f/yx

To find the stationary points of f(x, y), we set f/x = 0 and f/y = 0 so that 2x – 4y² = 0 or x = 2y² …(1) and -8xy + 12y³ = 0 or, 2x = 3y² … (2)

The equations 1 and 2 have only one solution which is x = 0 and y = 0 so that we have only one stationery point ( 0 , 0). Now,   (²f/x² )( ²f/y² ) ²f/xy = 2(-8x + 36y² ) – ( - 8y) = - 16x + 8y + 72 y² = 0 at ( 0.0) . Thus ( 0 , 0) can be anything either a minima or a maxima.