CALCULUS I Chapter III Additionnal Applications of the Derivative Mr. Saâd BELKOUCH

 Increasing and Decreasing Functions; Relative Extrema  Concavity and Points of Inflection  Curve Sketching 2

Increasing and Decreasing Functions; Relative Extrema  Intuitively, we regard a function f(x) as increasing where the graph of f is rising, and decreasing where the graph is falling. 3

Increasing and decreasing functions  Let f(x) be a function defined on the interval a < x < b, and let X l and X 2 be two numbers in the interval. Then: f(x) is increasing on the interval if f(X 2 ) > f(X I ) whenever X 2 > X l f(x) is decreasing on the interval if f(X 2 ) X l 4

 As demonstrated in the figure below, if the graph of a function f(x) has tangent lines with only positive slopes on the interval a 0. Similarly, we can say the same for negative slopes. 5

Procedure for Using the Derivative to Determine Intervals of Increase and Decrease for a Function f  Step 1. Find all values of x for which f' (x) = 0 or f' (x) is not continuous, and mark these numbers on a number line. This divides the line into a number of open intervals.  Step 2. Choose a test number c from each interval a < x < b determined in step 1 and evaluate f'(c). Then, If f'(c) > 0, the function f(x) is increasing (graph rising) on a < x < b. If f’(c) < 0, the function f(x) is decreasing (graph falling) on a < x <b. 6

 Ex: Intervals of increase and decrease for: f(x) = 2x 3 + 3x x - 7 f'(x) = 6x 2 + 6x - 12 = 6(x + 2)(x - 1) which is continuous everywhere, with f'(x) = 0 where x = 1 and x = -2  The numbers -2 and 1 divide the x axis into three open intervals; namely, x 1. Choose a test number c from each of these intervals; say, c = -3 from x 1. Then evaluate f'(c) for each test number: f'(-3) = 24> 0 f'(0) = We conclude that f' (x) > 0 for x 1, so f(x) is increasing (graph ris­ing) on these intervals. Similarly, f'(x) < 0 on - 2 < x < 1, so f(x) is decreasing (graph falling) on this interval. 7

 NOTATION We shall indicate an interval where f(x) is increasing by an "up arrow" and an interval where f(x) is decreasing by a "down arrow“. Thus, the results in the last example can be represented by this arrow, diagram: 8

Relative Extrema  A "peak" on the graph of a function f is known as a relative maximum, and a "valley" is a relative minimum. Thus, a relative maximum is a point on the graph of f that is at least as high as any nearby point on the graph, while a relative minimum is at least as low as any nearby point. 9

10 Since a function f(x) is increasing when f'(x) > 0 and decreasing when f'(x) < 0, the only points where f(x) can have a relative extremum are where f'(x) =0 or f'(x) does not exist. Such points are so important that we give them a special name

First Derivative Test for Relative Extrema 11

A procedure for sketching the graph of a continuous function using its derivative 12

2. Concavity and points of inflection  If the function f(x) is differentiable on the interval a < x < b, then the graph of f is concave upward on a < x < b if the slope of the tangent lines to the graph is increasing on the interval, and concave downward on a < x < b if it is decreasing on the interval 13

Determining Intervals of Concavity  we observed that a function f(x) is increasing where its derivative is positive. Thus, the derivative func­tion f'(x) must be increasing where its derivative f’’(x) is positive. Suppose f’’(x) > 0 on an interval a < x < b. Then f'(x) is increasing, which in turn means that the graph of f(x) is concave upward on this interval. Similarly, on an interval a < x < b, where f’’(x) < 0, the derivative f'(x) will be decreasing and the graph of f(x) will be con­cave downward. 14

 Ex:intervals of concavity for f(x) = 2x 6 - 5x 4 + 7x – 3 f'(x) = 12x x and f"(x) = 60x x 2 = 60r(x 2 - 1) = 60x(x - 1)(x + 1) The second derivative f"(x) is continuous for all x and f"(x) = 0 for x = 0, x = 1, and x = -1. These numbers divide the x axis into four intervals on which I" (x) does not change sign; namely, x 1. Evaluating f"(x) 15

 Thus, the graph of f(x) is concave up for x 1 and concave down for -1 < x < 0 and for 0 < x < I, as indicated in this concavity diagram.  The graph of f(x) will look like 16

End of Chapter 3 17