INCREASING AND DECREASING FUNCTIONS

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2 f (x) = x 3 x 2 2x increasing Walking uphill means the function is increasing

3 f (x) = x 3 x 2 2x increasing decreasing Walking uphill means the function is increasing Walking downhill means the function is deceasing

4 f (x) = x 3 x 2 2x increasing decreasing increasing Walking uphill means the function is increasing Walking downhill means the function is deceasing

5 f (x) = x 3 x 2 2x increasing decreasing increasing Walking uphill means the function is increasing Walking downhill means the function is deceasing Neither increasing nor decreasing at a peak or valley.

6 Definition Let f be a function defined on some interval, and let x 1 and x 2 be any real numbers in the interval with x 1 < x 2. We say that f is increasing on the interval if f (x 1 ) < f (x 2 ) We say that f is decreasing on the interval if f (x 1 ) > f (x 2 )

7 Below is the graph of the function f (x). Determine the interval(s) in which f (x) is increasing and decreasing. increasing: ( 3, 0) [ (2, 1) decreasing:( 1, 3) [ (0, 2)

8 Consider the previous graph. 4 slope > increasing decreasing increasing positive slope (f 0 (x) > 0) the function is increasing.

9 Consider the previous graph. slope = increasing decreasing increasing positive slope (f 0 (x) > 0) the function is increasing. slope of zero (f 0 (x) =0) the function is neither increasing nor decreasing.

10 THE DERIVATIVE TEST Consider the previous graph. 4 slope < increasing decreasing increasing positive slope (f 0 (x) > 0) the function is increasing. slope of zero (f 0 (x) =0) the function is neither increasing nor decreasing. negative slope (f 0 (x) < 0) the function is decreasing.

11 Consider the previous graph slope = increasing decreasing increasing positive slope (f 0 (x) > 0) the function is increasing. slope of zero (f 0 (x) =0) the function is neither increasing nor decreasing. negative slope (f 0 (x) < 0) the function is decreasing.

12 Consider the previous graph. 4 slope > increasing decreasing increasing positive slope (f 0 (x) > 0) the function is increasing. slope of zero (f 0 (x) =0) the function is neither increasing nor decreasing. negative slope (f 0 (x) < 0) the function is decreasing.

13 Derivative Test for Increasing and Decreasing Suppose f (x) is a function whose derivative exists at every point in some interval. if f 0 (x) > 0 for all x in the interval, the function is increasing on the interval. if f 0 (x) < 0 for all x in the interval, the function is decreasing on the interval. if f 0 (x) =0 for all x in the interval, the function is constant on the interval. How can we determine these intervals from the equation of the function?

14 If f 0 (x) goes from positive to negative (or vice verse) at a point x = a, then one of two things must be true: (1) f 0 (c) =0, or (2) f 0 (x) does not exist at x = c We call these points critical points. We can use the critical points of a function to determine the intervals over which it is increasing and decreasing.

15 Let f (x) = x x 2. Determine the open intervals where the f is inc./dec. Step 1: Find the critical points of f To do this, we first find f 0 (x): f 0 (x) = 3 Next, we determine for which x-values either f 0 (x) =0 or f 0 (x) is undefined. 3 2x = 0 ) 3 = 2x ) x = 3 2 Any other critical point would be where f 0 (x) =3 2x is undefined. However, f 0 (x) is a polynomial so it is defined for all real numbers, so... Critical point: x = 3 2 2x

16 Let f (x) = x x 2. Determine the open intervals where the f is inc./dec. Step 2: Use the derivative test From Step 1 we know that we have a critical point at 3/2 Our one critical point divides the real line into two intervals Choose one point a from the first interval and check if f 0 (a) > 0 or f 0 (a) < 0. f 0 ( 1) = 3 2( 1) = 5 > 0 So our original function f is increasing on the first interval f is increasing on (-,1.5) 0 1.5

17 Let f (x) = x x 2. Determine the open intervals where the f is inc./dec. Step 2: Use the derivative test Next, choose a point in the second interval f 0 (2) = 3 2(2) = 1 < 0 So our original function f is decreasing on the first interval f is decreasing on (1.5, ) Putting this all together: f is increasing: ( 1, 1.5) f is decreasing: (1.5, 1)

18 Determine the open intervals where the following functions are increasing and decreasing (1) f (x) =x 2/3 (2) g(t) = 3t + 6 (3) h(x) = p x 2 + 1

19 A county realty group estimates that the number of housing starts per year over the next three years will be H(r) = r 2 where r is the mortgage rate (in percent): (a) Where is H(r) increasing? (b) Where is H(r) decreasing?