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14.2 Extremal Values on a Curve in Two Dimensions

Suppose we have a curve, C that is defined by an equation, G ( x , y ) = 0 , and we seek an extremal value of F ( x , y ) among points restricted to lie on this curve.

Imagine, for example that G represents an ellipse, a x 2 + b y 2 = 1 , and we want the maximum of x y on C .

At any point q on C , we are free to move while staying on C only in the direction of the tangent line to the curve. Our condition above for an extremum then tells us that for q to be an extremum of F , F must have 0 derivative in the direction of the tangent, t ⟶ , to the curve defined by G .

This means that the gradient of F must be perpendicular to t ⟶ . But the gradient of G is perpendicular to t ⟶ as well, so that in two dimensions the gradient of F and the gradient of G must be parallel, for F to have an extremum on G .

There are two standard ways to express this condition.

One is to notice that it means that the parallelogram formed by ∇ ⟶ F and ∇ ⟶ G has no area, so that the determinant with these vectors as columns must be 0.

The other is to notice that it means that ∇ ⟶ F = c ∇ ⟶ G for some constant c .

Either observation allows us to find the extrema.

Example

The second method is called that of "Lagrange Multipliers", and the constant c is called a Lagrange Multiplier.

In the example you can click on above, if you write out the three equations defined by G = a x 2 + b y 2 − 1 = 0 , i ^ · ( ∇ ⟶ F − c ∇ ⟶ G ) = 0 and j ^ · ( ∇ ⟶ F − c ∇ ⟶ G ) = 0 , you may solve them for x , y and c , and arrive at the same solutions obtained.

Again, computing second derivatives (or examining values of F ) must be used to determine the local and/or global maxima and minima.

When a curve is defined parametrically with parameter t you can write F = F ( x ( t ) , y ( t ) ) and apply the single variable condition that d F d t = 0 .

Exercises:

14.1 Work out the details of the Lagrange Multipliers approach to the example above.

14.2 Suppose we want to maximize the volume of a vertically oriented cylinder given a fixed value q for the surface area of its sides and its top (but not its bottom). What radius and height should it have?

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