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1 Unit 1. Calculus Topic 4: Increasing and decreasing functions: turning points In topic 4 we continue with straightforward derivatives and integrals: Locate turning points where f () = 0. Determine the nature of turning points. Increasing and decreasing functions A function = f() is increasing on a set of values if, as the -values increase, the function s -values increase. A function = f() is decreasing on a set of values if, as the -values increase, the function s -values decrease. Eample A The graph of = f () is shown. At the points A (where = 1) and B (where = 8), the curve is neither increasing nor decreasing. The values = 1 and = 8 are called stationar values of the function. 6 A 5 4 = f() B As the -values increase from 4 to 1, the -values increase from 1 to 5. The function = f () is said to be increasing on the interval 4 < < 1. As the -values increase from 1 to 8, the -values decrease from 5 to 1. The function = f() is said to be decreasing on the interval 1 < < 8. As the -values increase from 8, the function = f() is again an increasing function. Turning points A point at which a function changes from increasing to decreasing or vice versa is called a turning point (abbreviation TP). Turning points are stationar values of the function. If the function changes from increasing to decreasing, the turning point is called a local maimum. In Eample A, the point A(1, 5) is a local maimum because the -value at = 1 is the maimum value of the function for all -values close to 1. Note: When the -value eceeds about 11, the value of the function becomes greater than 5. Local minimums are turning points where the function changes from being a decreasing function to being an increasing function. In Eample A, the point B (8, 1) is a local minimum.

2 178 Unit 1. Calculus Tangents to graphs In the diagram below, several tangents to the curve = g() are drawn. = g () function increasing for < 1; tangents have positive gradients. function decreasing for 1 < < ; tangents have negative gradients. function increasing for > ; tangents have positive gradients. An tangent drawn at a point on the graph of a function which is increasing is tilted up and hence has a positive gradient. Thus at points on the graph of a function which is increasing, the derivative will be positive. Similarl when the function is decreasing the derivative is negative. This can be written: If = f () is an increasing function then f'() > 0. Similarl, if = f () is decreasing then f'() < 0. At turning points, tangents are horizontal and have a gradient of 0. At turning points, f'() = 0. turning points Determining the nature of turning points A graph will show clearl whether a turning point is a local maimum or minimum. Eample B Q. Find the turning point(s) of = and determine their nature. A. ' = 4 8 [differentiating = 8 + 4] At TP, 4 8 = 0 [' = 0 at the turning point(s)] = [solving for ] At =, = [substituting = in = 8 + 4] = 4 coordinates of the TP are (, 4). = is a quadratic function and its graph is a parabola The -intercept is (0, 4) and from the diagram, (, 4) is a local minimum Oford Universit Press

3 Topic 4: Increasing and decreasing functions: turning points 179 Eample C Q. 1. Find the turning points of = and determine their nature.. For what set of values is this function decreasing? A. ' = 8 [differentiating ] 8 = 0 ( 4)( + ) = 0 Oford Universit Press ' = 0 at an TP = or 4 [factorising] when =, = 1 ( ) ( ) 8( ) + [substituting = in ] = = 1 TP is (, 1 1 ) when = 4, 1 = [substituting = 4 in ] = TP is (4, ) 40 Substituting = 0 in = (, shows that the function ) has a -intercept of (0, ). 10 = A sketch of the graph shows that (, 1 1 ) 10 0 is a local maimum and (4, ) is a 0 local minimum. 40 (4, ). The function is decreasing for all values in the interval < < 4. Another technique for determining the nature of turning points involves evaluating the gradient function in the neighbourhood of the turning point. Eample D Q. Find the turning points of = and determine their nature. A. From Eample C, the turning points occur at = and = 4. The table shows the values of the gradient function in the neighbourhoods of these two -values f () [for eample f' ( ) = ( ) ( ) 8 = 7 (positive)] Between = and = 1, the graph Between = and = 5, the graph changes from increasing (positive changes from decreasing (negative gradient) to decreasing (negative gradient) to increasing (positive gradient), so = is a maimum. gradient), so = 4 is a minimum.

4 180 Unit 1. Calculus 1 So (, 1 ) is a maimum, and (4, ) is a minimum turning point for the curve = Note: The nature of the turning points can also be determined b eamining the values of the function in the neighbourhood of the turning points. For eample, the table below shows that there is a minimum value at = f () Unit 1. Activit 4A: Turning points For each of the following functions, find the coordinates of the turning points and determine their nature. State the values for which the functions are increasing and those for which the functions are decreasing. 1. = = 6 8. = = = = = = = 10. a. State whether the curve = 5 4 is increasing, decreasing or stationar when: i. = ii. = iii. = 1 1 iv. = 0 b. Find the nature of an turning points. 11. f is a real number function such that f() = ( + 1)( 1) = + 1. a. Sketch a graph of f, marking the intercepts on the aes. b. Differentiate to find f'(). c. Find the values of that give turning points of f, stating the nature of these turning points. d. For what set of values of is f a decreasing function? e. Find f(0). f. Find f'(0). g. Find the equation of the tangent to the graph of f, at = = A + B + C is a quadratic function. Its derivative, ', has a value of when = 1 and a value of when =. In addition, the point (1, 1) lies on the graph of this function. Find A, B and C. Oford Universit Press

5 + + + Topic 4: Increasing and decreasing functions: turning points 181 Alternative tests for turning points The following are alternative tests for distinguishing maimum and minimum points of a graph and stationar points of inflection. In particular, these tests should be applied to the point = a if it is found that f'(a) = f'(a) = 0 (ie test failure occurs). The sign of the gradient is considered in the neighbourhood of the stationar point. Test for a maimum point If A is a maimum point at = a, then the gradient of the graph changes from positive (+) to negative ( ) as increases, as shown in the figure. The test for a maimum point is made b using a numerical value for just less than a and one for just greater than a, and substituting each value into the derivative f'(). These values for the gradients are entered in the table alongside to verif the maimum point at = a. Test for a minimum point If B is a minimum point at = b, then the gradient of the graph changes from negative ( ) to positive (+) as increases, as shown in the figure. The test for a minimum point is made using a numerical value for just less than b, and one for just greater than b, and substituting each value into the derivative f'(). These values for the gradients are entered in the table alongside to verif the minimum point at = b. < a = a > a f'() f '() > 0 f '(a) = 0 f '() < 0 maimum 0 < a a > a Sign of gradient Test for a stationar point of inflection If C is a stationar point of inflection at = c, then the gradient of the graph stas positive (or negative) on both sides of the stationar point, as increases A B < b = b > b f'() f '() < 0 f '(b) = 0 f '() > 0 minimum 0 < b b > b Sign of gradient C < c c > c Sign of gradient (+) C 0 0 < c c > c Sign of gradient ( ) < c = c > c f'() f '() > 0 f '(c) = 0 f '() > 0 < c = c > c f'() f '() < 0 f '(c) = 0 f '() < 0 stationar point of inflection stationar point of inflection Oford Universit Press

6 18 Unit 1. Calculus Polnomials A polnomial graph will have equation = a n n + a n 1 n a + a + a 1 + a 0. It is useful to have some idea of the graphs of polnomials. The following table shows eamples that could represent some polnomials. Degree Graph Description (quadratic) At most, one turning point. No points of inflection. 0 (cubic) At most, two turning points. One point of inflection. 0 4 (quartic) At most, three turning points. Two points of inflection. 0 Unit 1. Activit 4B: Curve sketching 1. Sketch a continuous curve = f () with these properties: a. f() =, f'() = 0, f() is concave down for all. b. f'() = f'(5) = 0, f(5) < 0 and f() = 0. c. f () = f'() = f''() = 0, f() is concave up for > and f() is concave down for <.. Find the coordinates of the turning points of f () = f() = has three turning points. Find their coordinates and identif which are maimum points and which are minimum points. Show full working. 4. The equation of a curve is = ( )( + ). a. State the -intercepts and the -intercept. b. Using calculus, find the -coordinates of the maimum and minimum points and distinguish between them using suitable tests. c. Sketch the curve showing the main features. 5. = ( )( + ) has two turning points. a. Find their coordinates and identif which is which, using suitable tests. b. Identif the -intercepts and -intercept. c. Sketch the curve showing its properties. Oford Universit Press

7 Topic 4: Increasing and decreasing functions: turning points The equation of a curve is f() = ( + )( ). a. Find the - and -intercepts of the graph and eplain how ou obtained our answers. b. Find the coordinates of the maimum and minimum points. c. Sketch the curve. 7. Find and identif the stationar points of the function f() = Find the coordinates of the turning points for = ( 4) The figure shows the graph of f () = ( ). The dotted line AB is a tangent to the curve at B, and = 0 is a tangent to the curve at the origin. Calculate the coordinates of A and B (do not do a scale drawing). 10. The equation of a polnomial curve of degree 4 is = 4 4. a. Find the coordinates of all the stationar points and distinguish between them. Justif our answers. 4 b. Eplain wh the -intercepts are 0 and. c. Find the non-stationar point of inflection. Justif our answer. d. Sketch the curve showing the main features. e. Write down the values of where the curve is concave down. 11. Find the coordinates of the stationar points for the function f () = 4 9 and show tests for distinguishing the maimum, minimum and points of inflection. Find all inflection points, the intercepts, and hence sketch the curve. 1. a. Find and identif the stationar points of the function f () = 5. If the curve crosses the -ais at = a, = b, = c, find a, b and c. Hence, sketch the curve. b. From the graph of f () = 5 find all the values of for which: i. f () is decreasing. ii. f() is concave up. 1. Investigate the turning points for the curve = 4 + p. a. For what values of p does this curve have three turning points? b. If p = 1 find the turning points and sketch the curve. A f() 0 B Oford Universit Press