OPTIMISATION PROBLEMS
Analysis
 

EXERCISE 3

The total costs of producing x units of a certain product is C(x)=9-2x+x3/6  and each unit is sold for (12-3x) of the unit of currency.

a) How many units need to be produced to give the minimum average cost per unit. 

  • Write the function in order to find the minimum, f(x)=C(x)/x

  • Find f'(x) and solve the equation: f'(x)=0

  • Find f''(x) for the obtained values

Change the value of x in the window. The graphs of f, f' and f'' will appear for you to check your results.

  • Where does f'(x) cut the X-axis?

  • What is the behaviour of f''(x) at these points?

b) Find the number of units that need to be sold to give the highest profit.

  • Write the function y=g(x)
    "profit = income - cost"

  • Find g'(x) and solve the equation: g'(x)=0

  • Find g''(x) and its sign at these values

As above, change the value of x in the window to check your results.

       
           
  María José García Cebrian
 
Spanish Ministry of Education. Year 2001
 
 

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