14.2 Extremal Values on a Curve in Two Dimensions

]> 14.2 Extremal Values on a Curve in Two Dimensions

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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, $\overset{⟶}{t}$ , to the curve defined by $G$ .

This means that the gradient of $F$ must be perpendicular to $\overset{⟶}{t}$ . But the gradient of $G$ is perpendicular to $\overset{⟶}{t}$ 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 $\overset{⟶}{\nabla} F$ and $\overset{⟶}{\nabla} G$ has no area, so that the determinant with these vectors as columns must be 0.

The other is to notice that it means that $\overset{⟶}{\nabla} F = c \overset{⟶}{\nabla} G$ 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 $G = a x^{2} + b y^{2} - 1 = 0, \hat{i} \cdot (\overset{⟶}{\nabla} F - c \overset{⟶}{\nabla} G) = 0$ and $\hat{j} \cdot (\overset{⟶}{\nabla} F - c \overset{⟶}{\nabla} G) = 0$ , 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 $\frac{d F}{d t} = 0$ .

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?