But it's often difficult or even impossible to write down parametrizations to implicitly-defined curves. The curve in the following problem is an example. That's why the method of Lagrange multipliers is so powerful -- you don't have to have a parametrization of the constraint equation.
f(x,y)=x^2+x*(y+2)when x and y are constrained to lie on the implicitly-defined curve
x^4-2*x^2+y^2=1/4.In other words, use graphical techniques to locate all points where grad f(x,y) is parallel to grad g(x,y). Use these points to find the maximum and minimum values of f(x,y) restricted to the curve.
Hint: Use the Maple command
implicitplot(x^4-2*x^2+y^2=1/4, x=-1.5..1.5, y=-1.5..1.5);
to plot the constraint curve in the (x,y) plane.
In your writeup include a plot of the graph of f(x,y) with a sketch of the restriction of f(x,y) to the constraint curve. Indicate the extreme points on this curve that you found with the above techniques.