Chapter 5 Applications of the Derivative Sections 5. 1, 5. 2, 5 Chapter 5 Applications of the Derivative Sections 5.1, 5.2, 5.3, and 5.4

Applications of the Derivative Maxima and Minima Applications of Maxima and Minima The Second Derivative - Analyzing Graphs

Absolute Extrema Let f be a function defined on a domain D Absolute Maximum Absolute Minimum

Absolute Extrema A function f has an absolute (global) maximum at x = c if f (x)  f (c) for all x in the domain D of f. The number f (c) is called the absolute maximum value of f in D Absolute Maximum

Absolute Extrema A function f has an absolute (global) minimum at x = c if f (c)  f (x) for all x in the domain D of f. The number f (c) is called the absolute minimum value of f in D Absolute Minimum

Generic Example

Generic Example

Generic Example

Relative Extrema A function f has a relative (local) maximum at x  c if there exists an open interval (r, s) containing c such that f (x)  f (c) for all r  x  s. Relative Maxima

Relative Extrema A function f has a relative (local) minimum at x  c if there exists an open interval (r, s) containing c such that f (c)  f (x) for all r  x  s. Relative Minima

Generic Example The corresponding values of x are called Critical Points of f

Critical Points of f (stationary point) (singular point) A critical number of a function f is a number c in the domain of f such that (stationary point) (singular point)

Candidates for Relative Extrema Stationary points: any x such that x is in the domain of f and f ' (x)  0. Singular points: any x such that x is in the domain of f and f ' (x)  undefined Remark: notice that not every critical number correspond to a local maximum or local minimum. We use “local extrema” to refer to either a max or a min.

Fermat’s Theorem If a function f has a local maximum or minimum at c, then c is a critical number of f Notice that the theorem does not say that at every critical number the function has a local maximum or local minimum

Generic Example Two critical points of f that do not correspond to local extrema

Example Find all the critical numbers of Stationary points: Singular points:

Graph of Local max. Local min.

Extreme Value Theorem If a function f is continuous on a closed interval [a, b], then f attains an absolute maximum and absolute minimum on [a, b]. Each extremum occurs at a critical number or at an endpoint. a b a b a b Attains max. and min. Attains min. but no max. No min. and no max. Open Interval Not continuous

Finding absolute extrema on [a , b] Find all critical numbers for f (x) in (a , b). Evaluate f (x) for all critical numbers in (a , b). Evaluate f (x) for the endpoints a and b of the interval [a , b]. The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a , b], and the smallest value found is the absolute minimum for f on [a , b].

Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Absolute Max. Absolute Min. Evaluate Absolute Max.

Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Absolute Max. Absolute Min.

Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Max. Evaluate Absolute Min.

Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Max. Absolute Min.

Increasing/Decreasing/Constant

Increasing/Decreasing/Constant

Increasing/Decreasing/Constant

The First Derivative Test

Generic Example A similar Observation Applies at a Local Max.

The First Derivative Test Determine the sign of the derivative of f to the left and right of the critical point. left right conclusion f (c) is a relative maximum f (c) is a relative minimum No change No relative extremum

Relative Extrema Example: Find all the relative extrema of Stationary points: Singular points: None

The First Derivative Test Find all the relative extrema of Relative max. f (0) = 1 Relative min. f (4) = -31 + 0 - 0 + 0 4

The First Derivative Test

The First Derivative Test

Another Example Find all the relative extrema of Stationary points: Singular points:

Stationary points: Singular points: Relative max. Relative min. + ND + 0 - ND - 0 + ND + -1 0 1

Graph of Local max. Local min.

Domain Not a Closed Interval Example: Find the absolute extrema of Notice that the interval is not closed. Look graphically: Absolute Max. (3, 1)

Optimization Problems Identify the unknown(s). Draw and label a diagram as needed. Identify the objective function. The quantity to be minimized or maximized. Identify the constraints. 4. State the optimization problem. 5. Eliminate extra variables. 6. Find the absolute maximum (minimum) of the objective function.

Optimization - Examples An open box is formed by cutting identical squares from each corner of a 4 in. by 4 in. sheet of paper. Find the dimensions of the box that will yield the maximum volume. x 4 – 2x x x x 4 – 2x

Critical points: The dimensions are 8/3 in. by 8/3 in. by 2/3 in. giving a maximum box volume of V  4.74 in3.

Optimization - Examples An metal can with volume 60 in3 is to be constructed in the shape of a right circular cylinder. If the cost of the material for the side is $0.05/in.2 and the cost of the material for the top and bottom is $0.03/in.2 Find the dimensions of the can that will minimize the cost. top and bottom cost side

So Sub. in for h

Graph of cost function to verify absolute minimum: 2.5 So with a radius ≈ 2.52 in. and height ≈ 3.02 in. the cost is minimized at ≈ $3.58.

Second Derivative

Second Derivative - Example

Second Derivative

Second Derivative

Concavity Let f be a differentiable function on (a, b). 1. f is concave upward on (a, b) if f ' is increasing on aa(a, b). That is f ''(x)  0 for each value of x in (a, b). 2. f is concave downward on (a, b) if f ' is decreasing on (a, b). That is f ''(x)  0 for each value of x in (a, b). concave upward concave downward

Inflection Point A point on the graph of f at which f is continuous and concavity changes is called an inflection point. To search for inflection points, find any point, c in the domain where f ''(x)  0 or f ''(x) is undefined. If f '' changes sign from the left to the right of c, then (c, f (c)) is an inflection point of f.

Example: Inflection Points Find all inflection points of

Inflection point at x  2 - 0 + 2

The Point of Diminishing Returns If the function represents the total sales of a particular object, t months after being introduced, find the point of diminishing returns. S concave up on S concave down on The point of diminishing returns is at 20 months (the rate at which units are sold starts to drop).

The Point of Diminishing Returns S concave down on Inflection point S concave up on