Use Lagrange multipliers to find the maximum and minimum values of?

The object function using Lagrange multiplier is

h(x,y) = 4 x + y + m (x^2 + 4 y^2 - 1)

dh/dx = 4 + 2 m x

dh/dy = 1 + 8 m y

dh/dm = x^2 + 4 y^2 - 1

dh/dx = 0, and dh/dy = 0 imply m = - 2 / x = -1 / (8y), therefore -16 y = - x ==> x = 16 y

Substituting this in the equation of the ellipse (16 y)^2 + 4 y^2 = 1

260 y^2 = 1, Therefore y = +/- 1/sqrt(260), and x = 16/sqrt(260)

The maximum value is 4(16/sqrt(260)) + 1/sqrt(260) = 65 / sqrt(260)

And minimum value is -65/ sqrt(260)

Note that the contours of the function are the straight lines 4x + y = c, The maximum and minimum

occur at the values of (x,y) where the slope is equal to (-4) which is the slope of the contour.

Differentiating implicity, 2x + 8 y dy/dx = 0 , i.e. dy/dx = - x /(4y),

Hence we want -x /(4y) = -4, i.e. x = 16 y, and the solution continues as before from this point.