Homework 9 Problems
Summary Constrained Optimization
You have an objective function f(x1, x2, . . . , xn) to minimize (or maximize), subject to
constraint of the form g(x1, . . . , xn) = c; the latter is called an equality constraint. At a
maximum or minimum it must be the case that
∇f=λ∇g
for some constant λ(the Lagrange multiplier). This is really nequations in n+ 1 unknowns;
the equation g=cis the (n+ 1)st equation. Solve these equations for the coordinates
(x1, . . . , xn). Plug each solution into the objective function fand pick out the max or min.
Inequality Constraints
If we have an inequality constraint g(x1, . . . , xn)≤cthen here’s what we do: At the max
value either g=cor g < c. In the former case we can use Lagrange Multipliers to pick out
the max or min of fon g=c. But if the max/min occurs at some point pwhere g < c (so the
max/min occurs AWAY from the constraint curve) then it must be the case that ∇f(p) = 0.
The reason is that if ∇f(p) isn’t zero, then there is some uphill (or downhill) direction from
p, so we can move a bit in that direction (since we’re away from the constraint) and so
increase or decrease the value of f.
This gives a “recipe” for solving max/min problems for objective function fwith inequal-
ity constraint g≤c:
1. Find all critical points pat which ∇f(p) = 0and g(p)< c.
2. Use Lagrange multipliers to find all solutions to ∇f(p) = λ∇g(p), g(p) = c.
3. Among all the points pfound in parts (1) and (2), choose that which yields the largest
or smallest value for f(p), as appropriate.
Problems
1. Find the maximum and minimum values of f(x, y) = x+xy subject to x2+y2/3≤1.
Just for fun, plot the constraint curve (use implicitplot) and the gradient of f(use
fieldplot), to verify your conclusion.
2. Among all points on the line 3x+ 5y= 7 in the plane, which one is closest to the
origin?
3. Which point on the curve x3+x2y= 3 is closest to origin?
4. Find three non-negative numbers x, y, z whose sum is 6 and product as large as possible.
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