Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates of the ordered pairs steadil increase, the -coordinates of the ordered pairs along the particle s path first decrease and then increase. Determining the intervals in which a function increases and decreases is etremel useful for understanding the behaviour of the function. The following statements give a clear picture: = Intervals of Increase and Decrease We sa that a function f is decreasing on an interval if, for an value of 1 6 on the interval, f 1 1 7 f 1. Similarl, we sa that a function f is increasing on an interval if, for an value of on the interval, f 1 1 6 f 1. 1 6 For the parabola with the equation, the change from decreasing -values to increasing -values occurs at 1,, the verte of the parabola. The function f 1 is decreasing on the interval 6 and is increasing on the interval 7. If we eamine tangents to the parabola anwhere on the interval where the -values are decreasing (that is, on 6 ), we see that all of these tangents have negative slopes. Similarl, the slopes of tangents to the graph on the interval where the -values are increasing are all positive. tangent CHAPTER 165
For functions that are both continuous and differentiable, we can determine intervals of increasing and decreasing -values using the derivative of the function. In the case of,. For 6, and the slopes of the tangents d d 6, are negative. The interval 6 corresponds to the decreasing portion of the graph of the parabola. For 7, and the slopes of the tangents are positive on d 7, the interval where the graph is increasing. We summarize this as follows: For a continuous and differentiable function, f, the function values (-values) are increasing for all -values where f 1 7, and the function values (-values) are decreasing for all -values where f 1 6. EXAMPLE 1 Using the derivative to reason about intervals of increase and decrease Use our calculator to graph the following functions. Use the graph to estimate the values of for which the function values (-values) are increasing, and the values of for which the -values are decreasing. Verif our estimates with an algebraic solution. a. 3 3 b. 1 Solution a. Using a calculator, we obtain the graph of 3 3. Using the TRACE ke on the calculator, we estimate that the function values are increasing on 6, decreasing on 6 6, and increasing again on 7. To verif these estimates with an algebraic solution, we consider the slopes of the tangents. The slope of a general tangent to the graph of 3 3 is given b We first determine the values of for which. These d 3 6. d values tell us where the function has a local maimum or local minimum value. These are greatest and least values respectivel of a function in relation to its neighbouring values. Setting, d we obtain 3 6 31, These values of locate points on the graph where the slope of the tangent is zero (that is, where the tangent is horizontal). 166.1 INCREASING AND DECREASING FUNCTIONS
Since this is a polnomial function it is continuous so is defined for all values d of. Because onl at and, the derivative must be either d positive or negative for all other values of. We consider the intervals 6, 6 6, and 7. Value of 6 6 6 7 Sign of 3( ) d d 7 d 6 d 7 Slope of Tangents positive negative positive Values of Increasing or Decreasing increasing decreasing increasing So 3 3 is increasing on the intervals 6 and 7 and is decreasing on the interval 6 6. b. Using a calculator, we obtain the graph of Using the TRACE ke 1. on the calculator, we estimate that the function values (-values) are decreasing on 6 1, increasing on 1 6 6 1, and decreasing again on 7 1. We analze the intervals of increasing/decreasing -values for the function b determining where is positive and where it is negative. d 1 (Epress as a product) 1 1 1 d 11 1 1 1 11 1 1 1 1 1 1 1 1 1 1 Setting, d 1 1 1 1 we obtain 1 or 1 1 1 1 (Product and chain rules) (Simplif) (Solve) CHAPTER 167
These values of locate the points on the graph where the slope of the tangent is. Since the denominator of this rational function can never be, this function is continuous so is defined for all values of. Because at 1and 1, d d we consider the intervals 1 q, 1, 1 1, 1, and 11, q. Value of 1 q, 1 1 1, 1 11, q Sign of d 1 1 1 d 6 d 7 d 6 Slope of Tangents negative positive negative Values of Increasing or Decreasing decreasing increasing decreasing Then is increasing on the interval 1 1, 1 and is decreasing on the 1 intervals 1 q, 1 and 11, q. EXAMPLE 8 = f () 6 6 8 = f() 6 8 Graphing a function given the graph of the derivative Consider the graph of f 1. Graph f 1. 8 = f () 6 6 6 Solution When the derivative, f 1, is positive, the graph of f 1 is rising. When the derivative is negative, the graph is falling. In this eample, the derivative changes sign from positive to negative at.6. This indicates that the graph of f 1 changes from increasing to decreasing, resulting in a local maimum for this value of. The derivative changes sign from negative to positive at.9, indicating the graph of f 1 changes from decreasing to increasing resulting in a local minimum for this value of. One possible graph of f 1 is shown. 168.1 INCREASING AND DECREASING FUNCTIONS
IN SUMMARY Ke Ideas A function f is increasing on an interval if, for an value of 1 6 in the interval, f 1 1 6 f 1. A function f is decreasing on an interval if, for an value of 1 6 in the interval, f 1 1 7 f 1. f( 1 ) f( ) f( 1 ) f( ) a 1 b a 1 b For a function f that is continuous and differentiable on an interval I f1 is increasing on I if f 1 7 for all values of in I f1 is decreasing on I if f 1 6 for all values of in I Need to Know A function increases on an interval if the graph rises from left to right. A function decreases on an interval if the graph falls from left to right. The slope of the tangent at a point on a section of a curve that is increasing is alwas positive. The slope of the tangent at a point on a section of a curve that is decreasing is alwas negative. Eercise.1 K C PART A 1. Determine the points at which f 1 for each of the following functions: a. f 1 3 6 1 c. f 1 1 1 1 9 b. f 1 d. f 1 5 1. Eplain how ou would determine when a function is increasing or decreasing. 3. For each of the following graphs, state i. the intervals where the function is increasing ii. the intervals where the function is decreasing iii. the points where the tangent to the function is horizontal CHAPTER 169
a. c. b. d.. Use a calculator to graph each of the following functions. Inspect the graph to estimate where the function is increasing and where it is decreasing. Verif our estimates with algebraic solutions. a. f 1 3 3 1 d. f 1 1 3 b. f 1 5 5 1 e. f 1 3 3 1 c. f 1 1 f. f 1 1 A PART B 5. Suppose that f is a differentiable function with the derivative f 1 1 1 1 1 3. Determine the values of for which the function f is increasing and the values of for which the function is decreasing. 6. Sketch a graph of a function that is differentiable on the interval 5 and that satisfies the following conditions: The graph of f passes through the points 1 1, and 1, 5. The function f is decreasing on 6 6 1, increasing on 1 6 6, and decreasing again on 6 6 5. 7. Find constants a, b, and c such that the graph of f 1 3 a b c will increase to the point 1 3, 18, decrease to the point 11, 1, and then continue increasing. 8. Sketch a graph of a function f that is differentiable and that satisfies the following conditions: f 1 7, when 6 5 f 1 6, when 5 6 6 1 and when 7 1 f 1 5 and f 11 f 1 5 6 and f 11 17.1 INCREASING AND DECREASING FUNCTIONS
9. Each of the following graphs represents the derivative function f 1 of a function f 1. Determine i. the intervals where f 1 is increasing ii. the intervals where f 1 is decreasing iii. the -coordinate for all local etrema of f 1 Assuming that f 1, make a rough sketch of the graph of each function. a. c. =f'() =f'() b. d. =f'() =f'() 1. Use the derivative to show that the graph of the quadratic function f 1 a b c, a 7, is decreasing on the interval 6 b and increasing on the interval 7 b a a. 11. For f 1 3, find where f 1, the intervals on which the function increases and decreases, and all the local etrema. Use graphing technolog to verif our results. 1. Sketch a graph of the function g that is differentiable on the interval 5, decreases on 6 6 3, and increases elsewhere on the domain. The absolute maimum of g is 7, and the absolute minimum is 3. The graph of g has local etrema at 1, and 13, 1. T PART C 13. Let f and g be continuous and differentiable functions on the interval a b. If f and g are both increasing on a b, and if f 1 7 and g1 7 on a b, show that the product fg is also increasing on a b. 1. Let f and g be continuous and differentiable functions on the interval a b. If f and g are both increasing on a b, and if f 1 6 and g1 6 on a b, is the product fg increasing on a b, decreasing, or neither? CHAPTER 171