Relating the Graphs of f, f’ and f’’ Section 4.3 Relating the Graphs of f, f’ and f’’

First Derivative Test for Local Extrema TBT: What is a critical point of a function? Point at which f’ is zero or undefined. At a critical point c….. 1. If f’ changes sign from positive to negative at c, then f has a local maximum value at c.

First Derivative Test 2. If f’ changes sign from negative to positive at c, then f has a local minimum at c. 3. If f’ does not change sign at c (same sign on both sides), then f has no local extreme value at c.

How can you tell if an endpoint is a max or min numerically? Analyze the slope near the endpoints.

Examples (a) Find local max and mins. (b) Identify the intervals in which the function is increasing or decreasing. 1. f(x) = x3 – 6x2 + 9x + 1 2. 𝑓 𝑥 = ln 𝑥 𝑥 3

The Second Derivative What does y’’ tell you about the graph of y’? When y’’ is positive, y’ is increasing. When y’’ is negative, y’ is decreasing. What does y’’ tell you about the graph of y? When y’’ is positive, the slope of y is increasing. When y’’ is negative, the slope of y is decreasing. What does increasing/decreasing slope look like? When y’’ is positive, y is concave up. When y’’ is negative, y is concave down.

Concavity on a Graph Where does the concavity change on the graph below? These points are called points of inflection and they are critical points for y’’.

Examples 3. Use the function from example 1: f(x) = x3 – 6x2 + 9x + 1 Determine where this function is concave up and concave down. Combine the information learned from f’’ with the info learned from f’ into one chart. Sketch the graph of f(x) using the information in the chart.

Additional Example f(x) = x2ex 1. Find the x-coordinates for any critical points. 2. Determine where f is increasing/decreasing. 3. Determine the max and mins. 4. Find where the function is concave up/down. 5. Locate any points of inflection. 6. Make a rough sketch based on your answers from 1-5.

Application to Motion Position = s(t) Velocity = v(t) = s’(t) Acceleration = a(t) = v’(t) = s’’(t) Remember that these are all vectors. Positive values generally indicate to the right or up. Negative values usually refer to left or down. Speed is not a vector. If acceleration and velocity have the same sign, the object is speeding up (either in a + or – direction. If a(t) and v(t) have opposite signs, the object is slowing down.

Example s(t) = 3t4 – 16t3 + 24t2 1. When is the object moving left/right? 2. When does the object reverse direction? 3. When is the velocity increasing/decreasing? 4. When is the object speeding up or slowing down? 5. Describe the motion of the object (including its speed) in words 6. Sketch a graph of the position curve. 7. Sketch a graph of the velocity curve. 8. Verify answers using the graphing calculator.

Examining Critical Values Suppose y’ = 𝑥 𝑦 . What can you tell me about the point (0, 4)? Why can’t you determine if it is a max or a min? We don’t have other values for y’ at nearby points. Second Derivative Test for Local Extrema If f’(c) = 0 and f’’(c) < 0, then f has a local maximum at x = c. If f’(c) = 0 and f’’(c) > 0, then f has a local minimum at x = c.