1. C2: Maxima and Minima Problems
Learning Objective: to use differentiation to find maximum and minimum points and solve problems set in a practical context

2. Starter: Find the stationary points on the curve
y = 2x3- 15x2 + 24x + 6 and determine, by finding the second derivative, whether the stationary points are maximum, minimum or points of inflexion.
Find the greatest value of 6x - x2. State the range of the function f(x) = 6x - x2

3. Examples: A closed rectangular box with a square base has a total surface area of 6m2. Find its greatest possible volume.

4. Sketch the curve y = x2 – x + 3

5. A farmer wishes to fence in a rectangular enclosure of area 200m2
A farmer wishes to fence in a rectangular enclosure of area 200m2. One side of the enclosure is formed by part of a wall already in position. What is the least possible length of fencing required for the other three sides?

6. A square of side x cm is cut from each of the corners of a rectangular piece of cardboard 15cm x 24cm. The cardboard is then folded to form an open box of depth x cm. Show that the volume of the box is (4x3 – 78x x) cm3. Find the value of x for which the volume is a maximum.

7. A cylindrical can is made so that the sum of its height and the circumference of its base is 45π cm. Find the radius of the base of the cylinder if the volume of the can is a maximum. r h

8. Sketch the graph of the curve y = x2 – 3x

9. Task 1: C2 Book – Exercise 9C, page 150