CHAPTER 3 SECTION 3.4 CONCAVITY AND THE SECOND DERIVATIVE TEST

Definition of Concavity and Figure 3.24

Sketch 4 graphs a)1 decreasing and concave up b)1 increasing and concave up, c)1 decreasing and concave down, d)1 increasing and concave down a b c d xx xx yy y y

Concave upward Look at these two graphs. Each is concave upward, but one is decreasing and the other is increasing. We need to be able to determine concavity from the function and not just from the graph. For each of the graphs above sketch the tangent lines to the graph at a number of different points. xx yy

As we move from left to right the slopes of the tangent lines are getting less negative. That is they are increasing. x y

As we move from left to right the slopes of the tangent lines are getting larger. That is they are increasing.

When a graph is concave upward The slope of the tangent lines are increasing.

Concave downward Look at these two graphs. Each is concave downward, but one is decreasing and the other is increasing. We need to be able to determine concavity from the function and not just from the graph. For each of the graphs above sketch the tangent lines to the graph at a number of different points. x x yy

As we move from left to right, the slopes of the tangent lines are getting more negative. They are decreasing.

As we move from left to right the slopes of the tangent lines are getting smaller. That is they are decreasing.

When a graph is concave downward The slopes of the tangent lines are decreasing.

Putting it all together For a function f that is differentiable on an interval I, the graph of f is (i) Concave up on I, if the slope of the tangent line is increasing on I or (ii) Concave down on I, if the slope of the tangent line is decreasing on I

Linking knowledge (i) Concave up on I, if the slope of the tangent line is increasing on I. If the slope of the tangent line is increasing and the slope of the tangent line is represented by the first derivative and to determine when something is increasing we had to take the derivative, then to find where the slope of the tangent line (f ‘(x)) is increasing we will need to take the derivative of f ‘(x) or find the second derivative f “(x) I know, this is a very large run on sentence.

Linking knowledge (ii) Concave down on I, if the slope of the tangent line is decreasing on I If the slope of the tangent line is decreasing and the slope of the tangent line is represented by the first derivative and to determine when something is decreasing we had to take the derivative, then to find where the slope of the tangent line (f ‘(x)) is decreasing we will need to take the derivative of f ‘(x) or find the second derivative f “(x)

Definition of concavity For a function f that is differentiable on an interval I, the graph of f is (i) Concave up on I, if f’ is increasing on I or (ii) Concave down on I, if f’ is decreasing on I

Theorem 3.7 Test for concavity

Putting it all together Given the function f(x) f(x) = 0x-intercepts f(x) undefinedvertical asymptote f(x)>0Q-1 or Q-2 f(x)<0Q-3 or Q-4

Putting it all together

Determining concavity Determine the open intervals on which the graph is concave upward or concave downward. Concavity find second derivative. Find hypercritical numbers. Set up a chart Find concavity

c = 1; c = -1 and f” is defined on the entire # line

Setting up the chart intervalTest points Sign of f”f ‘concave (-∞, -1)-2+incupward (-1,1)0-dec downward (1,∞)2+incupward

Points of inflection A point of inflection for the graph of f is that point where the concavity changes.

Theorem 3.7 Test for Concavity

Definition of Point of Inflection and Figure 3.28

Theorem 3.8 Points of Inflection

Theorem 3.9 Second Derivative Test and Figure 3.31

Example 1: Graph the function f given by and find the relative extrema. 1 st find graph the function.

Example 1 (continued): 2 nd solve f (x) = 0. Thus, x = –3 and x = 1 are critical values.

Example 1 (continued): 3 rd use the Second Derivative Test with – 3 and 1. Lastly, find the values of f (x) at –3 and 1. So, (–3, 14) is a relative maximum and (1, –18) is a relative minimum.

Second Derivative Test a.If f’’(c) > 0 then ________________________ If c is a critical number of f’(x) and… b.If f’’(c) < 0 then ________________________ c.If f’’(c) = 0 or undefined then __________________________________

Second Derivative Test a.If f’’(c) > 0 then ________________________ (c, f(c)) is a relative min If c is a critical number of f’(x) and… b.If f’’(c) < 0 then ________________________ c.If f’’(c) = 0 or undefined then __________________________________ (c, f(c)) is a relative max the test fails (use 1 st Derivative test)

The second derivative gives the same information about the first derivative that the first derivative gives about the original function. If f’’(x) > 0 ______________ If f’’(x) < 0 ______________ If f’’(x) = 0 ______________ Concave upward Concave downward Inflection Points If f’(x) > 0 ______________ If f’(x) < 0 ______________ If f’(x) = 0 ______________ ______________ ________________ For f(x) to increase, _____________ For f’(x) to increase, _____________ For f(x) to decrease, _____________ For f’(x) to decrease, _____________

The second derivative gives the same information about the first derivative that the first derivative gives about the original function. If f’’(x) > 0 ______________ If f’’(x) < 0 ______________ If f’’(x) = 0 ______________ Concave upward Slopes increase Concave downward Slopes decrease f(x) increases f(x) decreases f(x) is constant Inflection Points Where concavity changes Occur at critical numbers of f”(x) If f’(x) > 0 ______________ If f’(x) < 0 ______________ If f’(x) = 0 ______________ f’(x) decreases f’(x) increases f(x) is conc up f(x) is conc down f’(x) is constant f(x) is a straight line ______________ ________________ For f(x) to increase, _____________ For f’(x) to increase, _____________ For f(x) to decrease, _____________ For f’(x) to decrease, _____________ f’(x) > 0 f’’(x) > 0 f’(x) < 0 f’’(x) < 0

Sketch Include extrema, inflection points, and intervals of concavity.

Sketch Critical numbers: No VA’s smooth None Include extrema, inflection points, and intervals of concavity. Critical numbers: None

1.Find the extrema of

Crit numbers: rel min at rel max at rel min at rel max at 2 nd Derivative Test

2.Sketch

Crit numbers: rel min at rel max at 2 nd Derivative Test Crit numbers: Intervals: Test values: Inf pt f ’’(test pt) f(x)

2 -2 2

Find a Function Describe the function at the point x=3 based on the following: 3 (3, 4)

Find a Function Describe the function at the point x=5 based on the following: 5

Find a Function Given the function is continuous at the point x=2, sketch a graph based on the following: 2 (2,3)

WHY? BECAUSE f ’ (x) is POSITVE!!!!!!!!!!!!!!!