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Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad , Glen Whiffen , John Owen, Robert Haese , Sandra Haese and Mark Bruce Haese and Haese Publications, 2004. Practice. Worksheet S-47 #3 y = x 3 + 1.5 x 2 – 6 x – 3

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Material Taken From:Mathematicsfor the international student Mathematical Studies SLMal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark BruceHaese and Haese Publications, 2004


  • Worksheet S-47 #3

  • y = x3 + 1.5x2 – 6x – 3

    Find where the gradient is equal to zero.

Maximum vs minimum
Maximum vs. Minimum

  • If f ’(p)=0, then p is a max or min.

    • p is a maximum if f ’(x) is ________ to the left of p and ________ to the right of p.

    • p is a minimum if f ’(x) is ________ to the left of p and ________to the right of p.






if f ’(x) is positive then f(x) is ___________.

if f ’(x) is negative then f(x) is ___________.



Worksheet S-47 #4, 5


Section 19JK - Optimization

  • At a maximum or minimum  tangent line is horizontal  derivative is zero.

  • We can use that information to find the maximum and minimum of a real-world situation.


Example 1

A sheet of thin card 50 cm by 100 cm has a square of side x cm cut away from each corner and the sides folded up to make a rectangular open box.

Find the volume, V, of the box in terms of x.

Using calculus, find the value of x which gives a maximum volume of the box.

Find this maximum volume.

See animation in HL book, page 653


Example 2

A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length x cm are cut from each corner of the card as shown in the diagram below. What is left is then folded to make an open box, of length l cm and width w cm.

  • (a) Write expressions, in terms of x, for

  • (i) the length, l;

  • (ii) the width, w.

  • (b) Show that the volume (B m3) of the box is given by B = 4x3 – 66x2 + 216x.

  • (c) Find .

  • (d) (i) Find the value of x which gives the maximum volume of the box.

  • (ii) Calculate the maximum volume of the box.


Example 3

A rectangle has width x cm and length y cm. It has a constant area 20 cm2.

  • Write down an equation involving x, y and 20.

  • Express the perimeter, P, in terms of x only.

  • Find the value of x which makes the perimeter a minimum and find this minimum perimeter.


Example 4

An open rectangular box is made from thin cardboard. The base is 2x cm long and x cm wide and the volume is 50 cm3. Let the height be h cm.

Write down an equation involving 50, x, and h.

Show that the area, y cm2, of cardboard used is given by y = 2x2 + 150x – 1

Find the value of x that makes the area a minimum and find the minimum area of cardboard used.

See animation in HL book, page 653


  • Worksheet S-47 #6, 7

  • Pg 629 #5,6,7

  • Worksheet, Optimization