1. Chapter 11 Maximum and minimum points and optimisation problems Learning objectives:  Understand what is meant by stationary point  Find maximum and minimum points  Find second derivatives  Use the second derivative to determine the nature of a stationary point  Solve practical optimisation problems using differentiation

2. 1. Maximum and minimum points

3. EG4. A curve has equation y = x 3 + 3x 3 - 9x - 21 a) Find the coordinates of the two stationary points of the curve. b) Find the second derivative and, hence, determine the nature of the stationary points. c) Hence, sketch the curve.

4. 2. Stationary points of inflection EG4. Find the stationary points on the curve with equation y = x 3 (4 - x) and determine their nature.

5. 3. Optimisation EG5: A farmer wants to build a rectangular pen for some animals and has 100m of fencing. One side of the pen is to be an existing wall. Find the dimensions of the pen enclosing the maximum area. EG6: The sum of two positive integers is 12. Find the two numbers such that the product of one number and the cube of the other number is as large as possible.

6. EG7. The diagram below shows a rectangle sheet of metal 10cm by 16cm. A square of side x cm is cut from each corner and the metal is then folded to make an open box (i.e. the box does not have a lid). a) Show that the Volume, Vcm 3, the box can hold is given by V = 4x 3 - 52x 2 + 160x b) Differentiate V with respect to x and hence find the values of x for which dy/dx = 0. c) Show that there is a single value of x for which V is a stationary. Verify that this value of x gives a maximum value of V. d) Calculate the maximum possible volume of the box.