Second Derivatives, Inflection Points and Concavity

Important Terms turning point: points where the direction of the function changes maximum: the highest point on a function minimum: the lowest point on a function local vs absolute: a max can be a highest point in the entire domain (absolute) or only over a specified region within the domain (local). Likewise for a minimum. increase: the part of the domain (the interval) where the function values are getting larger as the independent variable gets higher; if f(x1) < f(x2) when x1 < x2; the graph of the function is going up to the right (or down to the left) decrease: the part of the domain (the interval) where the function values are getting smaller as the independent variable gets higher; if f(x1) > f(x2) when x1 < x2; the graph of the function is going up to the left (or down to the right) "end behaviour": describing the function values (or appearance of the graph) as x values getting infinitely large positively or infinitely large negatively or approaching an asymptote

New Term – Graphs Showing Concavity

New Term – Concave Up Concavity is best “defined” with graphs (i) “concave up” means in simple terms that the “direction of opening” is upward or the curve is “cupped upward” An alternative way to describe it is to visualize where you would draw the tangent lines  you would have to draw the tangent lines “underneath” the curve

New Term – Concave down Concavity is best “defined” with graphs (ii) “concave down” means in simple terms that the “direction of opening” is downward or the curve is “cupped downward” An alternative way to describe it is to visualize where you would draw the tangent lines  you would have to draw the tangent lines “above” the curve

New Term – Concavity In keeping with the idea of concavity and the drawn tangent lines, if a curve is concave up and we were to draw a number of tangent lines and determine their slopes, we would see that the values of the tangent slopes increases (become more positive) as our x-value at which we drew the tangent slopes increase This idea of the “increase of the tangent slope is illustrated on the next slides:

New Term – Concave Up

New Term – Concave Down

Functions and Their Derivatives In order to “see” the connection between a graph of a function and the graph of its derivative, we will use graphing technology to generate graphs of functions and simultaneously generate a graph of its derivative Then we will connect concepts like max/min, increase/decrease, concavities on the original function to what we see on the graph of its derivative

Example 1

Example 1 Points to note: (1) the fn has a minimum at x=2 and the derivative has an x-intercept at x=2 (2) the fn decreases on (-∞,2) and the derivative has negative values on (-∞,2) (3) the fn increases on (2,+∞) and the derivative has positive values on (2,+∞) (4) the fn changes from decrease to increase at the min while the derivative values change from negative to positive (5) the function is concave up and the derivative fn is an increasing fn (6) the second derivative graph is positive on the entire domain

Example 2

Example 2 f(x) has a max. at x = -3.1 and f `(x) has an x-intercept at x = -3.1 f(x) has a min. at x = -0.2 and f `(x) has a root at –0.2 f(x) increases on (-, -3.1) & (-0.2, ) and on the same intervals, f `(x) has positive values f(x) decreases on (-3.1, -0.2) and on the same interval, f `(x) has negative values At the max (x = -3.1), the fcn changes from being an increasing fcn to a decreasing fcn  the derivative changes from positive values to negative values At a the min (x = -0.2), the fcn changes from decreasing to increasing  the derivative changes from negative to positive f(x) is concave down on (-, -1.67) while f `(x) decreases on (-, -1.67) and the 2nd derivative is negative on (-, -1.67) f(x) is concave up on (-1.67,  ) while f `(x) increases on (-1.67, ) and the 2nd derivative is positive on (-1.67, ) The concavity of f(x) changes from CD to CU at x = -1.67, while the derivative has a min. at x = -1.67

Second Derivative – A Summary If f ‘’(x) >0, then f(x) is concave up If f ‘(x) < 0, then f(x) is concave down If f ‘’(x) = 0, then f(x) is neither concave nor concave down, but has an inflection points where the concavity is then changing directions The second derivative also gives information about the “extreme points” or “critical points” or max/mins on the original function: If f ‘(x) = 0 and f ‘’(x) > 0, then the critical point is a minimum point (picture y = x2 at x = 0) If f ‘(x) = 0 and f ‘’(x) < 0, then the critical point is a maximum point (picture y = -x2 at x = 0) These last two points form the basis of the “Second Derivative Test” which allows us to test for maximum and minimum values

Examples - Algebraically Find where the curve y = x3 - 3x2 - 9x - 5 is concave up and concave down. Find and classify all extreme points. Then use this info to sketch the curve. f(x) = x3 – 3x2 - 9x – 5 f `(x) = 3x2 – 6x - 9 = 3(x2 – 2x – 3) = 3(x – 3)(x + 1) So f(x) has critical points (or local/global extrema) at x = -1 and x = 3 f ``(x) = 6x – 6 = 6(x – 1) So at x = 1, f ``(x) = 0 and we have a change of concavity Then f ``(-1) = -12  the curve is concave down, so x = -1 must represent a maximum point Also f `(3) = +12  the curve is concave up, so x = 3 must represent a minimum point Then f(3) = -33, f(-1) = 0 as the ordered pairs for the function whose graph is shown on the next slide: