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Extremum problems: numbers and economics

Calculus, grade 11 (4 units) Lesson 13 of 20Learning

Extremum problems: numbers and economics

The way to a solution

An extremum problem asks for the greatest or the least value of a quantity – a sum, a cost, an area. The way, as in grade 10:

  • Choose a variable x\displaystyle x for one of the quantities, and write the others with it.
  • Write the quantity to make greatest or least as a function of x\displaystyle x.
  • Write the interval of x\displaystyle x: the values possible in the problem (a positive length, between 0 and 15 items …).
  • Differentiate, and solve f′(x)=0\displaystyle {f'(x)}={0} inside the interval.
  • Check the sign of f′\displaystyle f' on both sides of each solution: from −\displaystyle - to +\displaystyle + – a minimum, from +\displaystyle + to −\displaystyle - – a maximum.
  • Answer the question asked – sometimes it is about x\displaystyle x, sometimes about the value of the function or about another quantity.
An open and a closed interval

On an open interval (x>0\displaystyle x>0, 0<x<10\displaystyle 0<x<10) the ends are not in the interval, and the extremum is a point where f′=0\displaystyle {f'}={0}.
On a closed interval (5≤x≤15\displaystyle 5\leq x\leq15) the ends are in the interval too, and the greatest or least value may be at an end. So compare:

  • the values at the points where f′=0\displaystyle {f'}={0} inside the interval;
  • the values at the two ends.
Example: a number and a fraction

x\displaystyle x is a positive number.
What is the least value of the sum x+9x\displaystyle x+\frac{9}{x}?

In the solution below we differentiate S(x)=x+9x\displaystyle {S(x)}={x+\frac{9}{x}} on the interval x>0\displaystyle x>0, and check the sign of the derivative:

  • S′(x)=1−9x2=x2−9x2\displaystyle {S'(x)}={1-\frac{9}{x^2}}={\frac{x^2-9}{x^2}}.
  • S′(x)=0\displaystyle {S'(x)}={0} when x2=9\displaystyle {x^2}={9}; in the interval – only x=3\displaystyle {x}={3}.
  • When 0<x<3\displaystyle 0<x<3, S′<0\displaystyle S'<0; when x>3\displaystyle x>3, S′>0\displaystyle S'>0. So at x=3\displaystyle {x}={3} there is a minimum.
  • The least sum: S(3)=3+3=6\displaystyle {S(3)}={3+3}={6}.
Example: the average cost

Making x\displaystyle x items a day in a factory costs C(x)=x2+20x+400\displaystyle {C(x)}={x^2+20x+400} shekels.
The average cost of an item is the whole cost divided by the number of items: A(x)=C(x)x=x+20+400x\displaystyle {A(x)}={\frac{C(x)}{x}}={x+20+\frac{400}{x}}.
In the solution below we find how many items a day give the least average cost:

  • A′(x)=1−400x2\displaystyle {A'(x)}={1-\frac{400}{x^2}}; A′(x)=0\displaystyle {A'(x)}={0} when x2=400\displaystyle {x^2}={400}, and on the interval x>0\displaystyle x>0: x=20\displaystyle {x}={20}.
  • When x<20\displaystyle x<20, A′<0\displaystyle A'<0; when x>20\displaystyle x>20, A′>0\displaystyle A'>0: a minimum.
  • A(20)=20+20+20=60\displaystyle {A(20)}={20+20+20}={60}: an item costs 60 ₪ on average when 20 items are made a day.
A(x)=x+20+400x\displaystyle A(x)=x+20+\frac{400}{x}
M=(20,60)\displaystyle M=(20,60)
Exercise 1

x\displaystyle x is a positive number, and S(x)=x+1x\displaystyle {S(x)}={x+\frac{1}{x}} – the sum of x\displaystyle x and its reciprocal.

a

Differentiate S\displaystyle S.

S(x)=x+1x\displaystyle S(x)=x+\frac{1}{x}

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

    Solve the equation S′(x)=0\displaystyle {S'(x)}={0}.

      c

      For which positive number is the sum of the number and its reciprocal least? What is this sum?

      x\displaystyle x

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        The sum

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 2

          The product of two positive numbers is 50.
          The sum of the first number and twice the second is to be as small as possible.
          The first number is x\displaystyle x.

          a

          The second number is 50x\displaystyle \frac{50}{x}. Write the sum S(x)\displaystyle S(x) on the board.

          b

          Differentiate S\displaystyle S.

          S(x)=x+100x\displaystyle S(x)=x+\frac{100}{x}

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            c

            Solve the equation S′(x)=0\displaystyle {S'(x)}={0}.

              d

              Work out the least sum. What are the two numbers?

              S(x)=x+100x\displaystyle S(x)=x+\frac{100}{x}

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                The first number

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  The second number

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    Exercise 3

                    x\displaystyle x is a positive number.
                    For which x\displaystyle x is the fraction xx2+9\displaystyle \frac{x}{x^2+9} greatest?

                    a

                    Differentiate f(x)=xx2+9\displaystyle {f(x)}={\frac{x}{x^2+9}}.

                    f(x)=xx2+9\displaystyle f(x)=\frac{x}{x^{2}+9}

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                      b

                      Solve the equation f′(x)=0\displaystyle {f'(x)}={0}, and check that the positive solution is a maximum.

                        c

                        Work out the greatest value of the fraction.

                        f(x)=xx2+9\displaystyle f(x)=\frac{x}{x^{2}+9}

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                          Exercise 4

                          A workshop makes between 5 and 15 tables a week.
                          When it makes x\displaystyle x tables, the average cost of a table is A(x)=x2+10+200x\displaystyle {A(x)}={\frac{x}{2}+10+\frac{200}{x}} hundred shekels.

                          a

                          Differentiate A\displaystyle A.

                          A(x)=x2+10+200x\displaystyle A(x)=\frac{x}{2}+10+\frac{200}{x}

                            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                            b

                            Solve the equation A′(x)=0\displaystyle {A'(x)}={0}. Does it have a solution in the interval 5≤x≤15\displaystyle 5\leq x\leq15?

                              c

                              Work out the average cost at the two ends of the interval.

                              A(x)=x2+10+200x\displaystyle A(x)=\frac{x}{2}+10+\frac{200}{x}

                                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                                A(x)=x2+10+200x\displaystyle A(x)=\frac{x}{2}+10+\frac{200}{x}

                                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                                  d

                                  How many tables a week give the least average cost? And when is it greatest?

                                  For the least cost

                                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                                    For the greatest cost

                                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power