Home  18.013A  Chapter 14 


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 $xy$ 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, $\stackrel{\u27f6}{t}$ , to the curve defined by $G$ .
This means that the gradient of $F$ must be perpendicular to $\stackrel{\u27f6}{t}$ . But the gradient of $G$ is perpendicular to $\stackrel{\u27f6}{t}$ 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 $\stackrel{\u27f6}{\nabla}F$ and $\stackrel{\u27f6}{\nabla}G$ has no area, so that the determinant with these vectors as columns must be 0.
The other is to notice that it means that $\stackrel{\u27f6}{\nabla}F=c\stackrel{\u27f6}{\nabla}G$ for some constant $c$ .
Either observation allows us to find the extrema.
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 $G=a{x}^{2}+b{y}^{2}1=0,\widehat{i}\xb7(\stackrel{\u27f6}{\nabla}Fc\stackrel{\u27f6}{\nabla}G)=0$ and $\widehat{j}\xb7(\stackrel{\u27f6}{\nabla}Fc\stackrel{\u27f6}{\nabla}G)=0$ , 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 $\frac{dF}{dt}=0$ .
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?
