In lectures we did questions like (in the simplest form) f(x,y,z) = x + 2y + z subject to a constraint, eg. x^2 + y^2 + z^2 = 48 Find the critical points, value and determine the max/min
In the exam papers some of the questions have the constraint with <= instead of just =. Does this make a big difference to how I approach the question?
Also, when I get the critical values, how do I determine which points are a max/min? Not specifically the example above but in general.
Lastly does the term "on a closed disk" in place of constraint hold any significance?
Thanks in advance for any help...
I doubt that you did not cover this in class- just in a different section that you are thinking of. If you have, for example, the problem that says "minimize f(x,y,z) on you would probably, as ThePerfectHacker suggested, use Lagrange multipliers.
But if the problem is "minimize f(x,y,z) on , that's a completely different problem. You would first minimize f(x,y,z) on the interior of the sphere by finding [itex]\nabla f[/itex], finding where it is 0, and seeing if any such points are inside the sphere. Then you would look for additional max and min on the surface of the sphere- that is by using the constraint .
Actually, for the example you give, f(x,y,z)= x+ 2y+ z, is never 0 so the problem really reduces to the constraint on the surface of the sphere.
If you are talking about global min or max on a closed bounded set, just calculate the value of the function at each critical value and see which is largest and which smallest. For determining whether critical values are local max or min or neither, surely your text has the "second derivative test": "If [tex]D= f_xxf_yy- (f_xy)^2> 0 and either f_xx or f_yy is positive, the critical point is a minimum. If D> 0 and either f_xx or f_yy is negative, the critical point is a maximum. If D< 0 the critical point is a saddle point. If D= 0 this test doesn't work."Also, when I get the critical values, how do I determine which points are a max/min? Not specifically the example above but in general.
That is the same as the ball above. Look for critical points inside the disk and then look for additional critical points on the boundary of the disk.Lastly does the term "on a closed disk" in place of constraint hold any significance?
Thanks in advance for any help...