Optimization

 

1.  Read the problem

2.  Draw a diagram

3.  Define the variables and label

4.  Write equations based on the variables (called optimization equations).

5.      Differentiate – use the first and second derivative test to find the max/min.

 

Examples

 

1.  Find two numbers whose difference is 100 and whose product is a minimum.

     ßoptimization equation

    Write the optimization equation with one variable and use the first derivative to find the minimum.

Text Box: Because is always positive, -50 must be a minimum.           

           

 

 

 

Text Box: ß Final answerText Box: Now solve for a using the original equation.           

 

 

2. A 3 by 3 square is going to be turned into an open-topped box by cutting the corners and folding it upward.  Find the maximum volume that this box can have.

                                    v = lwh          ßoptimization equation   

 










Text Box: Draw a diagram


 

 


           

 

 

 

 

 

 

     

      Write the optimization equation with one variable and use the first derivative to find the minimum.

Text Box: .29 is a maximum because it goes from increasing to decreasingText Box: Use quadratic formula to solveText Box: Use the product rule to find            

Final answer à
 
     

 

 

 


  Maximize the volume of the cone.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Text Box: Find first derivative and use to solve for zeroes.
 

 

 

 

 

 

 

 

Text Box: The zeroes are 1 and 3
 

 

 

Text Box: Plug in the critical numbers found above into the second derivative. If the answer is negative, there is a maximum. 

 

 

 

 

 






Text Box: When x=1, is negative, therefore when x=1, the volume is at a maximum.



Text Box: First Derivative Test: C is a critical number on a continuous function F. 1. If for all and for all , then is the absolute maximum value of F. 2. If for all and for all then is the absolute minimum value of F.


Text Box: Second Derivative Test: C is a critical number on a continuous function F. 3. If when , then c is a minimum. 4. If when , then c is a maximum.