Calculus I (MAT 145) Dr. Day Monday Oct 30, 2017 Using Derivatives: Function Characteristics & Applications (Ch 4) Extreme Values (4.1) Determining Function Behavior from its Derivatives (4.3) Increasing/Decreasing Nature of a Function (first derivative) Concavity of a Function (second derivative) Monday, October 30, 2017 MAT 145

Absolute and Relative Extremes Absolute (Global) Extreme: An output of a function such that it is either the greatest (maximum) or the least (minimum) of all possible outputs. Relative (Local) Extreme: An output of a function such that it is either the greatest (maximum) or the least (minimum) in some small neighborhood along the x-axis. Extreme Value Theorem: For f(x) continuous on a closed interval, there must be extreme values. If f is continuous on a closed interval [a,b] then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b]. Fermat’s Theorem: If f has a local max or min at x = c and if f ’(c) exists then f ’(c) = 0. Critical Point: An interior point (not an endpoint) on f(x) with f ’(x) = 0 or f ’(x) undefined. Note: The function MUST EXIST at x = c for a critical point to exist at x = c. Monday, October 30, 2017 MAT 145

Absolute (global) Extremes and Relative (local) Extremes Monday, October 30, 2017 MAT 145

Absolute Extrema What is the absolute maximum in the graph? (This means “Tell me the greatest y-value.”) Where is the absolute maximum located? (This means, “Tell me the x-value that corresponds to the maximum y-value.”) What is the absolute minimum in the graph? Where is the absolute minimum located? What do you notice about the locations of absolute extrema? Where could they occur? Monday, October 30, 2017 MAT 145

Extreme Value Theorem: If f is continuous on the closed interval [a,b] then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b]. Monday, October 30, 2017 MAT 145

Relative (Local) Extrema In the graph, Relative (Local) Maxima are: f(b) (y-value) found at x=b f(d) (y-value) found at x=d These values are greater than all the other y-values in a small neighborhood immediately to the right and left. The Relative (Local) Minima are: f(c) (y-value) found at x=c f(e) (y-value) found at x=e These values are less than all the other y-values nearby. Monday, October 30, 2017 MAT 145

Relative (Local) Extrema Where could relative extrema occur? Critical numbers are the locations where local extrema could occur. Critical points are the points (x- and y-values) that describe both the locations and function values at those points. Determine critical numbers for 1. 2. 3. Monday, October 30, 2017 MAT 145

Fermat’s Theorem Monday, October 30, 2017 MAT 145

Absolute Extrema—Closed Interval Method Monday, October 30, 2017 MAT 145

Absolute Extrema Where and what are the absolute and local extrema? Monday, October 30, 2017 MAT 145

Absolute and Relative Extremes Ways to Find Extrema Local Extremes: examine behavior at critical points Absolute Extremes: examine behavior at critical points and at endpoints Example Determine critical numbers, absolute extrema, and relative extrema for the unrestricted function (all possible domain values) and then for the restricted domain [−1,3]. Monday, October 30, 2017 MAT 145

What does f’ tell us about f? If f’(c)= 0, there is a horizontal tangent to the curve at x=c. This may mean there is a local max or min at x=c. If f’(c) is undefined, there could be a discontinuity, a vertical tangent, or a cusp (sharp point) at x=c. If f(x) is continuous at x=c, there may be a local max or min at x=c. Monday, October 30, 2017 MAT 145

Absolute and Relative Maximums and Minimums Use the graph of f ’(x) to describe a strategy for identifying the global and local extrema of f, knowing f ’(x). m a t h Monday, October 30, 2017 MAT 145

First derivative test Monday, October 30, 2017 MAT 145

Absolute and Relative Maximums and Minimums Must every continuous function have critical points on a closed interval? Explain. Can an increasing function have a local max? Explain. Monday, October 30, 2017 MAT 145

Concavity of a Function Concavity Animations More Concavity Animations Monday, October 30, 2017 MAT 145

Concavity of a Function Concavity Animations More Concavity Animations Monday, October 30, 2017 MAT 145

What does f’’ tell us about f? If f’’(c)> 0, then the original curve f(x) is concave up at x=c. If f’’(c)< 0, then the original curve f(x) is concave down at x=c. If f’’(c)= 0, then f(x) is neither concave up nor concave down at x=c. And there could be an inflection point on f(x) at x=c. If f’’(c) is undefined, there could be a discontinuity, a vertical tangent, or a cusp (sharp point) in f’(x) at x=c. There may be a change of concavity in f(x) at x=c. Monday, October 30, 2017 MAT 145

Inflection Point Monday, October 30, 2017 MAT 145

Second derivative test Suppose f is continuous. What does it mean if f’(c)=0 and f’’(c)>0? What does it mean if f’(c)=0 and f’’(c)<0? Monday, October 30, 2017 MAT 145

First Derivative Test, Concavity, Second Derivative Test (Sec. 4.3) Determining Increasing or Decreasing Nature of a Function If f’(x) > 0, then f is _?_. If f’(x) < 0, then f is _?_. Using the First Derivative to Determine Whether an Extreme Value Exists: The First Derivative Test (and first derivative sign charts) If f’ changes from positive to negative at x=c, then f has a _?_ _?_ at c. If f’ changes from negative to positive at x=c, then f has a _?_ _?_ at c. If f’ does not change sign at x=c, then f has neither a local max or min at c. Concavity of f If f’’(x) > 0 for all x in some interval I, then the graph is concave up on I. If f’’(x) < 0 for all x in some interval I, then the graph is concave down on I. Second derivative Test If f’(c) = 0 and f’’(c) > 0, then f has a local min at c. If f’(c) = 0 and f’’(c) < 0, then f has a local max at c. Monday, October 30, 2017 MAT 145

Info about f from f ’ Here’s a graph of g’(x). Determine all intervals over which g is increasing and over which g is decreasing. Identify and justify where all local extremes occur. Monday, October 30, 2017 MAT 145

Info about f from f ’’ Here’s a graph of h”(x). Determine all intervals over which h is concave up and over which h is concave down. Identify and justify where all points of inflection occur. Monday, October 30, 2017 MAT 145

Pulling it all together For f(x) shown below, use calculus to determine and justify: All x-axis intervals for which f is increasing All x-axis intervals for which f is decreasing The location and value of every local & absolute extreme All x-axis intervals for which f is concave up All x-axis intervals for which f is concave down The location of every point of inflection. Monday, October 30, 2017 MAT 145

Monday, October 30, 2017 MAT 145

Identify Extrema From a Graph Graph each function. Identify all global and local extremes. For each of those, write a sentence based on this template: At x = ?, there is a (local/global) (max/min) of y = ? Monday, October 30, 2017 MAT 145

Identify Extrema From a Graph At x = 1 there is a global max of 5. At x = 4 there is a global min of 3. At x = 0 there is a global max of 4. There is no global min. At x = -1 there is a global min of 1/e. There is no global max. Monday, October 30, 2017 MAT 145

Determine the Critical Numbers For each function, determine every critical number. Unless otherwise restricted, assume that each function’s domain includes all possible values for which that function is defined. Monday, October 30, 2017 MAT 145

Determine the Critical Numbers For each function, determine every critical number. Unless otherwise restricted, assume that each function’s domain includes all possible values for which that function is defined. Monday, October 30, 2017 MAT 145

Identify Extrema Using Critical Numbers For each function, determine every critical number, and then use those critical numbers to determine all absolute extreme values. Note the domain restrictions. For each extreme value, write a sentence based on this template: At x = _?_, there is an absolute (max/min) of _?_. Monday, October 30, 2017 MAT 145

Identify Extrema Using Critical Numbers At x = √8 there is a global max of 8. At x = −1 there is a global min of −√15. At x = −2 there is a global max of 92. At x = 3 there is a global min of −158. At x = 1 there is a global max of 1. At x = 0 there is a global min of 0. Monday, October 30, 2017 MAT 145

Absolute and Relative Extremes Ways to Find Extrema Local Extremes: Examine behavior at critical points. Absolute Extremes: Examine behavior at critical points and at endpoints. Example Determine critical numbers, absolute extrema, and relative extrema for the unrestricted function (all possible domain values) and then for the restricted domain [−3,1]. Monday, October 30, 2017 MAT 145