A question for clarifcation concerning optimization

G

Guest

Guest
Hey, I was wondering. How do you maximize and minimize problem for an optimization problem? I've encountered these problems before during the school year, but I forgot how to do it. I am just reviewing for Calc BC so I won't be lost next yaer and well I just forgot considering it's the summer. lol. Unfortunately, can anybody help me clarify how do you minimize a problem and how to maxiumum a problem

For example for this certain problem... (out of the Larson Calc book 3-7 example 1)

A manufacturer wants to design an open box having a square base and a surface area of 108 sq in. What dimensions will produce a box with max volume? What dimensions (this is what I want to know; if you could max, can you also min?) will produce a box with a min volume?


Obviously, V=X^2H----
S=X^2+4XH=108
----
V=X^2H
=27X-X^3/4

0<X<RAD 108 FEASIBLE DOMAIN (what is a feasible domain?)

dv/dx=27-3x^2/4

3x^2=108

x=+/- 6
 
let x = edge of boxbase
let h be box height
Area of base =x^2
volume[V]=x^2h

surface area =x^2+4xh [base plus 4 sides] but surface area= 108 sq in.
108=x^2+4xh solve for h to substitute into equation for volume
h= [108-x^2] / 4x substitute
V= x^2[108-x^2] /4x or
V=x[108-x^2]/4 or
V=[108x -x^3] / 4
take derivative of V with respect to x, set it to 0 , to find max or min
dV/dx=[108 -3x^2]/4 set it =0
0=[108-3x^2]/4
0=108-3x^2
3x^2=108
x^2=36
x=+/- 6 -6 is extraneous answer out of reasonable domain
domain 0<x or
x=6 h= [108-36]/24
x=6 h=3 answer

is x= 6 a max or min? take 2nd derivative
d^2V/x^2=-3/2 x at x=6
d^2V/dX=-9 negative thus x= where V is maximum
if positiv,e, x=6 would bewhere V IS minimum

Arthur
 
All I am asking really is, for problems like this, if you could find the max, can you also find the min?
 
Max and Min depend on the nature of the problem.

If all your constraints are "less than or equal to" you probably are about finding a maximum and the minimum probably is zero (0).

If all your constraints are "greater than or equal to" you probably are about finding a minimum and the maximum probably is a very large number or there is none.
 
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