Question

How do you use lagrange multipliers to find the point (a,b) on the graph `y=e^(3x)` where the value ab is as small as possible?

Answer 1

You rewrite the problem in more useful form.

We usually express these problems as:
Minimize: (objective function)
Subject to: (constraint equation)

You want to minimize the product of two number, so the objective function is `f(x,y)=xy`.
The constraint is `y=e^(3x)`.

So the problem becomes:

Minimize: `f(x,y)=xy`
Subject to: `e^(3x)-y=0`

(If you prefer, you could use constraint `y-e^(3x)=0`)

Now, proceed as usual for a Lagrange multiplier problem.