4.3 Using Derivatives for Curve Sketching

Absolute maximum (also local maximum) Local maximum Local minimum Local extremes are also called relative extremes. Absolute minimum

Absolute maximum (also local maximum) Local maximum y’ DNE Local minimum Notice that local extremes in the interior of the function occur where is zero or is undefined.

Critical Point: A point in the domain of a function f at which or does not exist is a critical point of f . Note: Maximum and minimum points in the interior of a function always occur at critical points, but critical points are not always maximum or minimum values.

Critical points are not always extremes! (not an extreme)

(not an extreme) p

Point of Inflection A point in the domain of a function f at which or does not exist is a POSSIBLE point of inflection of f . Note: For a point to be an inflection point y’’ must change sign at that point. How do we find points of inflection using y’?

X- values where f’’ = 0 or DNE are not necessarily points of inflection! y = x4 y’’ = 0 but y does not change concavity at x = 0

X- values where f’’ = 0 or DNE are not necessarily points of inflection! y = x2/3 y’’ DNE at x = 0 but y does not change concavity at x = 0

A couple of somewhat obvious definitions: A function is increasing over an interval if the derivative is always positive. A function is decreasing over an interval if the derivative is always negative.

First derivative: is positive Function is increasing. is negative Function is decreasing. is zero or undefined Possible local maximum or minimum. Second derivative: is positive Curve is concave up. is negative Curve is concave down. is zero or undefined Possible inflection point (where concavity changes).

This is the graph of y = f’(x), the derivative of f(x). On what interval is f(x) increasing? (b) On what interval is f(x) decreasing?

This is the graph of y = f’(x), the derivative of f(x). (c) For what values of x does f have a relative maximum? Why? (d) For what values of x does f have a relative minimum? Why?

This is the graph of y = f’(x), the derivative of f(x). (e) On what intervals is f concave upward? Use f’ to explain. (f) On what intervals is f concave downward? Use f’ to explain.

This is the graph of y = f’(x), the derivative of f(x). (g) Find the x-coordinate of each point of inflection of the graph of f on the open interval (-3,5). Use f’ to justify your answer.

Stop here G 

First derivative test: Example: Graph There are roots at and . Possible extreme at . Set First derivative test: negative positive positive

First derivative test: Example: Graph There are roots at and . Possible extreme at . Set First derivative test: maximum at minimum at

Or you could use the second derivative test: Example: Graph There are roots at and . Possible extreme at . Set Or you could use the second derivative test: negative concave down local maximum positive concave up local minimum maximum at minimum at

Possible inflection point at . Example: Graph We then look for inflection points by setting the second derivative equal to zero. Possible inflection point at . negative positive inflection point at

p Make a summary table: rising, concave down local max falling, inflection point local min rising, concave up p

Let f(x) = x3 – 12x – 5. Use the first derivative test to find the local extreme values.

Let f’(x) = 4x3 – 12x2. Identify where the extrema of f occur. Find the intervals on which f is increasing and decreasing. Find where the graph of f is concave up and where it is Concave down. (d) Find any points of inflection.

Extreme Value Theorem: If f is continuous over a closed interval, then f has a maximum and minimum value over that interval. Maximum & minimum at interior points Maximum & minimum at endpoints Maximum at interior point, minimum at endpoint

Extreme values can be in the interior or the end points of a function. No Absolute Maximum Absolute Minimum

Absolute Maximum Absolute Minimum

Absolute Maximum No Minimum

No Maximum No Minimum

Finding Maximums and Minimums Analytically: 1 Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points. 2 Find the value of the function at each critical point. 3 Find values or slopes for points between the critical points to determine if the critical points are maximums or minimums. 4 For closed intervals, check the end points as well.

Finding Maximums and Minimums Analytically: Find the extrema of f(x) = 3x4 – 4x3 on the interval [-1,2]