4 3 How Derivatives Affect the Shape of

4. 3 How Derivatives Affect the Shape of a Graph: continuous differentiable f’(x) > 0 increasing f’(x) < 0 decreasing f’(x) = 0 constant

First find f’ and set = 0. Test the intervals around the critical values. Increase x<0 Decrease 0<x<1 Increase x>1 Short cut. Right Hand Behavior is always the same as the sign of the leading term of f’ and signs alternate to the left as long as each x-intercept is in numerical order and occurs an odd number of times. Even occurrences keeps the signs the same.

First find f’ and set = 0. Short cut. Right Hand Behavior is positive. x-int (3/2, 0) occurs once, so signs Decreasing interval. Increasing interval. alternate. x-int (0, 0) occurs twice, so signs DO NOT alternate. monotonic function a function that ALWAYS increases or ALWAYS decreases

The First Derivative Test: * if c is a critical # of f, + to – , then f has a relative _______ maximum 1. if f’ changes from ____ at c – to + minimum 2. if f’ changes from _____, then f has a relative _______ at c 3. if ____ no sign change, ___ no max or min

EX #2: Find the relative extrema. a. ) look back at EX #1 a… x = 0 is a max, + to – Local max at (0, 0) x = 1 is a min, – to + Local min at (1, -0. 5) b. ) look back at EX #1 b… x = 0 has no x = 3/2 is a min, sign change, – to + no max or min Local min at (3/2, -27/16)

EX #2: Find the relative extrema. Local min at Local max at Local min at

EX #3: a. ) On what intervals is f increasing? Decreasing? f’(x) > 0…Increasing interval. f’(x) < 0…Decreasing interval. b. ) At what values of x does f have local extrema? At x = 1, + to – Local max. At x = 5, – to + Local min. At x = 7, + to – Local max.

NOW, we are going to discuss where functions are curving upward or downward Concavity: * concave up * concave down Concavity Test: * if f is a function whose 2 nd derivative exists on an open interval, upward f ” (x) > 0 then the graph of f is concave _____ 1. if _____, f ” (x) < 0 then the graph of f is concave ______ 2. if _____, downward

Points of Inflection: where the graph changes concavity f ” (x) = 0 or f ”(x) _____ DNE * possible points of inflection are where ______ (as long as x is in the domain; ex: vertical asymptotes in any f / f ’ / f ”)

First find f’ and f ”. f is continuous. Points of Inflection (0, 0) and (2, -16) Local min at (3, -27) Concave UP DOWN Points of Inflection are at (0, 0) and (2, -16). f (x) will concave up to the left of (0, 0) and to the right of (2, -16). The graph will concave down between the inflection points

First find f’ and f ”. f is continuous. Concave UP Concave DOWN Concave UP

EX #5: Determine the intervals on which the graph is concave upward or downward. First find f’ and f ”. f is continuous. No points of inflection. Use the vert. asymptotes as boundaries to determine test points for concavity. Concave UP Concave DOWN Concave UP

2 nd Derivative Test: f’ (c) = 0 and _____, f ” (c) > 0 then f has a relative _______ min at x = c 1. if _____ f’ (c) = 0 and _____, f ” (c) < 0 then f has a relative _______ 2. if _____ max at x = c Relative min No extrema Relative max No extrema Relative min

Graphs: EX #7: Sketch the graph w/the given characteristics… f(0) = f(2) = 0 x-int at (0, 0) & (2, 0) f’(x) < 0 if x < 1 Decreasing left of x = 1 f’(1) = 0 Relative extrema at x = 1 f’(x) > 0 if x > 1 Increasing right of x = 1 f ”(1) > 0 Relative min at x = 1