1 Maxima and Minima The derivative measures the slope of the tangent to a curve. At the maximum or minimum points, the tangent is horizontal and has slope zero. Setting the derivative to zero, enables us to locate these max/min points. This process tells us where these max/min points are but not whether they are a maximum or minimum. So how can we tell ??

2 x y x1x1 x2x2 The first derivative would give us the value of x 1 and x 2 but not the fact that x 1 is a minimum point and x 2 is a maximum point. We can establish this by taking the second derivative, i.e. calculating the derivative of the derivative.

3 Test : If, at a turning point (max/min point) then the point is a minimum. If, the point is maximum.

4 Example: We know this has a maximum since Setting to find any turning points : turning point Since -8 < 0 then this point is a maximum.

5 Example: Turning points occur when Since ln x is undefined when x < 0, then is the only solution

6 When x = Since 4 > 0, the point is a minimum. Now,

7 Note: If, the point might not be a maximum or minimum. point of inflection

8 Solving Verbal Problems: (1) Read the question carefully. (2) Create an equation which gives a mathematical description of the process. (Sometimes given). (3) Calculate the derivative and find turning points. (4) Establish whether or not turning points are maximum or minimum or neither. (5) Interpret and explain the results in sentences.

9 Note : (3) If using the derivative does not produce the desired result, then the result will be one of the end points.

10 e.g. looking for the minimum value of some function A B C The first derivative will identify the maximum value (B) because it is a turning point. We are searching for the min value so check the value of the function at A and C. The lesser of the two gives the desired result.

11 Example: The cost per hour C (in dollars) of operating an automobile is given by where s is the speed in miles per hour. At what speed is the cost per hour a minimum?

12 Given Setting gives s = 50 is a turning point :. The turning point is a maximum.

13 Check endpoints of s. The minimum value occurs at s = 0. Costs are minimised when speed = 0 mi/hr.

14 Example: For a monopolist, the cost per unit of producing a product is $3 and the demand equation is What price will give the greatest profit?

15 Profit = total revenue - total costs

16 Setting

17 q = is a turning point :. q = is a maximum point The price that generates the maximum profit is A price of $6 for the product will produce maximum profit.

18 Example The cost of operating a truck on a throughway (excluding the salary of the driver) is 0.11+(s/300) dollars per mile, where s is the (steady) speed of the truck in miles per hour. The truck driver’s salary is $12 per hour. At what speed should the truck driver operate the truck to make a 700-mile trip most economical?

19 Cost per mile (1) :. For 700 miles, Driver’s salary = $12/hr, number of hours = Since distance = speed  time :. Cost of driver (2)

20 Total cost = (1) + (2)

21 Turning point

22 Checking with :. The turning point is a minimum. So costs are minimised when s = 60 mi/hr

23 A final note : It is worthwhile checking end points of the variables domain even when the calculus has produced a result. A B Calculus will reveal a maximum point at A but it can be seen the max value of the function occurs at B