Derivatives in Action Chapter 7

7.1 Introduction Derivatives measure slopes. A function’s maximum or minimum is often achieved at a stationary point where the slope is zero. Positive derivatives indicate an increasing slope, negative derivatives indicate a decreasing slope. Second derivatives allow to distinguish between “bowls and upended bowls.” Third derivatives help identify the inflection points. Differentials help approximate small changes in a function’s value.

7.2 Increasing and Decreasing Functions Since a function’s derivative is representing its slope, we have two handy rules for analyzing the function’s behavior in some set S in its domain: 1) If for , function is positively sloped for Consider the function : its derivative , so it is positive for every Hence, function has a positive slope for that range of x. 2) If , function is negatively sloped if For function its derivative is negative for x<0. Hence, this function has a negative slope for x<0

Strictly Increasing/Decreasing Functions Definition. A function whose slope is strictly positive in an interval [a,b] is called strictly increasing in that interval. Definition. A function whose slope is strictly negative in an interval [a,b] is called strictly decreasing in that interval. Definition. A function that is either strictly increasing or strictly decreasing in a specific interval is called strictly monotone in that interval.

Example 7.2 Consider a function . The derivative of this function is for all x. Hence, the slope of this function is everywhere negative, and the function is strictly decreasing for all x.

7.3 Optimization Definition. A function f(x) has a relative maximum at point J if the value of the function at J is strictly greater than the values of this function at all values of its argument in some vicinity around point J. Definition. A function f(x) has a relative minimum at point P if the value of the function at P is strictly less than the values of this function at all values of its argument in some vicinity around point P.

Terminology Optimization: quest for the best Maximization: find values in the domain of the function that result in its maximum value Minimization: find values in the domain of the function that result in its minimum value Extremum of a function: a function’s maximum or minimum An objective function: a function whose dependent variable is the object of optimization (minimization or maximization) Choice variables: the independent variables in the objective function whose values one can vary to arrive at an extremum

Stationary Points Definition. Consider a function f(x). If at point this function has a zero slope, i.e. if , the point is called a stationary point of function f(x). A function’s extremum is often a stationary point. However, the converse is not necessarily true: Point (0,0) is stationary, but is not an extremum. Both extrema are stationary points.

Optimization: An Example Suppose a firm’s profit depends on its revenues and costs in the following way: This firm’s objective is to maximize profits, so is the objective function, where profit is the object of maximization. The firm can alter the level of its profits by varying the amount of its output , so output is its choice variable. We shall later see that the optimal level of output for this firm will have to satisfy the following optimality condition: Optimum may not be unique!

Relative vs Absolute Extremum Optimization in its very general form deals with finding the maximum or minimum of some function, e.g. B y y B C A A B A D a b x a c d b x Both and result In the same value of function so we can consider both values as representing the function’s maximum (or minimum) While function may not have a finite extremum in the domain of real numbers, it does have two absolute extrema in [a,b] At b function reaches its absolute maximum in [a,b]. At x=c the function is reaching its local maximum in [a,b]. At point D the function is reaching both its local and global minimum.

Relative Extrema We agree to say that a function has a relative extremum at point if there exists a sufficiently small such that for every in case of a relative minimum, and in case of a relative maximum. y Function according to our definition reaches its local minimum at a b x

Definitions Definition. A point c in the domain D of function f is called a maximum point for f iff for all the value of function f at point c is such that Definition. Point c is a minimum point if in the definition above, Definition. In the definitions above, is called the maximum/minimum value. Definition. In case the inequality signs are strict, i.e. if in the definitions above or , we are talking about a strict maximum or a strict minimum. Collectively, we refer to optimal points and values, or extreme points and values.

7.4 A Maximum Value of a Function Theorem Suppose that a function f is differentiable in interval I and that c is an interior point of I. For x=c to be a maximum or a minimum point for function f in I, the first-order necessary condition must hold: FOC:

7.5 The Derivative as a Function of X Consider a function y=f(x). Suppose we can find its derivative everywhere for . Since is equal to the slope of function f(x) at various , we can talk about a mapping of into this function’s slopes at those . In other words, it is possible to define another function . In this way, a function’s derivative is also a function!

7.6 A Minimum Value of a Function

First Derivative Test At point C the function achieves its minimum according to the definition, but the slope of the tangency line is not zero: this is the case of the end point. The First Derivative Test: If function has an extreme value at some point , AND the derivative exists at that point, it is equal to zero: y A Both at A and B function reaches its relative extrema according to our definition, however, the derivative does not exist at either point. B C x

First Derivative Test: Formalization If the first derivative of a function at is , then the value of the function at will be: a. A maximum if the derivative changes its sign from positive to negative from the immediate left of the point to its immediate right. b. A minimum if the derivative changes its sign from negative to positive from the immediate left of the point to its immediate right. c. Neither a minimum nor a maximum if has the same sign on both the immediate left and the immediate right of point

A Graphical Illustration y x

A Numerical Example Find the relative extrema of the function 1. Find the derivative function: 2. To get the critical values, solve equation 3. Look at the sign of the derivative in the vicinity of is a relative minimum. 4. In the same fashion, in the vicinity of the derivative is changing its sign as well:

7.7, 7.8 Second Derivatives Definition. If function has the first derivative , the derivative of function in case it exists is called the second derivative, and is denoted by

Two Examples Higher order derivatives of the polynomial functions tend to vanish as the derivative order gets higher. Higher order derivatives do not always vanish

Curvature and Second Derivatives The value of the function is increasing (the slope > 0 ) The value of the function is decreasing (the slope < 0 ) The value of the slope of the function is increasing The value of the slope of the function is decreasing Consider the case of an increasing slope: y For negative numbers, since e.g. -2>-3, slope will be also increasing in the following case: C y Thus, second derivatives have something to do with the curvature of the function’s graph. B A x x

7.9 Worked Example Example 7.6 Consider a function . The slope of the graph of this function is given by its first derivative: The roots of this equation produce candidates for the minimum or maximum points. The second derivative will help us distinguish between a maximum and a minimum. Derivatives test does not help us locate the absolute minima or maxima.

Concave and Convex Functions Definition. A function is strictly convex if the part of the straight line connecting any two points A and B on the function’s graph lies above the graph. Points A and B are not taken into account. y In case of the strictly convex functions, for any two points on the function’s graph, the line segment connecting them will lie above the graph curve. The second derivative will be positive. A B x Definition. Similarly, a function is strictly concave if the part of the straight line connecting any two points A and B on the function’s graph lies below the graph. Points A and B are not taken into account. y B For the strictly concave functions a line segment connecting any two points on the function’s graph will lie below the graph curve. The second derivative will be negative. A x

Reverse Causality It is NOT correct to say that a function that is strictly convex/concave in some domain will necessarily have a positive/negative second derivative there. This function is strictly convex in the set of all real numbers, but at x=0 its second derivative is not negative: y It is typical for a strictly convex or concave function to have a positive or negative second derivative in most of their domains. x

An Application: Handling the Risk Toss the dice Consider the following game: Receive $20 if the number is even Receive $10 if the number is odd Your expected value of payoff will be In case you are charged $15 to enter the game, it’s called a fair bet. Will you play this game? Depends on your risk attitude represented by your utility function where x is the amount of money you have. U(x) U($15) is what you get when you DON’T play the game Thus, if you DO play the game, your expected payoff in terms of your utility is the simple average of U($20) and U($10) If you DO play and the number is even, you’re up for $5 more: you pay $15 to get in, you collect $20. Your end utility is U($20) The average utility is smaller than your utility when you don’t play the game, so you don’t play the game. Notice how this is related to the fact that the utility function is concave. If the number is odd, you lose $5 net, your end utility is U($10). $10 $20 $x $15

7.10 Points of Inflection Definition. Consider a function f(x) at some point , and its second derivative function . If the derivative exists, it is called the third derivative of function f(x), and is denoted by .

7.10 Points of Inflection Definition. A point such that 1) and 2) is called a point of inflection of function f(x). Definition. A point of inflection is called stationary inflection point if it is stationary. Minimum (maximum) slope interpretation. Since is the function’s slope at point , is the rate of increase of this slope. The third derivative is in fact the second derivative of the function f(x)’s slope function. In this way the two conditions and are sufficient conditions for a relative maximum or minimum of the slope function. This is why a point of inflection represents the point of a minimum or a maximum slope.

Example 7.7 Point K is an inflection point: the slope at K is zero, while it is positive on both sides of K. Or, at K the function has a minimum slope. To the left of K, e.g. at J, the tangent line lies above the graph, while to the right of K, e.g. at L, it lies below the graph. An inflection point can be thought of as the one where the tangent line lies on a different side of the function’s graph to the right and to the left of the inflection point. Two graphs at the bottom illustrate the fact that at K the slope is, indeed, a minimum slope.

Example 7.8 Point Q is the tangent point. At point Q, the tangent is above the graph to the right of Q, and it is below the graph to the left of Q. At point Q, the slope is at its maximum (middle graph). The second derivative is zero at Q, while the third derivative is negative. Alternatively, the second derivative of the slope function is negative, while the first derivative of the slope function is zero.

7.11 A Rule for Points of Inflection Stationary points that are points of inflection. Definition. Consider a function y=f(x). Point is called a stationary point of inflection if the following holds at that point: 1) 2) 3) In case at point , the slope of function f(x) is at its minimum in the vicinity of that point. In case at point , the slope of function f(x) is at its maximum in the vicinity of that point.

7.12 Non-Stationary Inflection Points The slope reaches its maximum at points P and M. The slope reaches its minimum at points Q and L.

Example 7.9 The curve is almost linear at point K, so it is not easy to see that K is an inflection point. At K, the slope of f(x) is at its maximum.

Example 7.10 M is an inflection point. At M, the slope is at its minimum (= most negative). The tangent line at M cuts the axis at point M.

Inflection Points between Extrema Proposition. If a function f(x) has a relative minimum and maximum in an interval [a,b], it must have at least one inflection point in [a,b]. Note: this proposition “works” for the functions for which derivatives exist up to the third order. Functions with discontinuities, for instance, are not subject to this proposition.

Stationary and Inflection Points

7.13 Convex and Concave Functions At points J and K the slope is positive since At point K the slope is greater compared to J. Remember that is the slope function, so an increasing slope means Definition. Consider a function f(x) in an interval [a,b]. If the slope of the graph of this function increases as x increases, the function f(x) is called convex from below in [a,b]. Functions that are convex from below in [a,b] have a positive second derivative in that interval:

Concave from Below Definition. Consider a function f(x) in an interval [a,b]. If the slope of the graph of this function decreases as x increases, the function f(x) is called concave from below in [a,b]. The slope is increasing in [0,5], but it is flatter at M compared to L. Alternatively, the slope function decreases in [0,5]. Functions that are concave from below in [a,b] have a negative second derivative in that interval:

Convex from Below: Decreasing Slope Convexity from below is defined in terms of the increasing slope irrespectively of the slope’s sign. The second derivative is positive in [0,5], so the slope is increasing in that interval even if the slope itself is negative. Tangents: if a tangent to the graph of a function lies below the graph, the function is convex from below.

Concave from Below: Decreasing Slope Convexity from below is defined in terms of the increasing slope irrespectively of the slope’s sign. The second derivative is positive in [0,5], so the slope is increasing in that interval even if the slope itself is negative. Tangents: if a tangent to the graph of a function lies above the graph, the function is concave from below.

Second Derivative Sign a Sufficient Condition Only Consider a function in [-1,1]. The second derivative is zero at x=0: However, the tangent line at x=0 lies below the function’s graph so the function is convex from below. Hence, or is a sufficient, not a necessary condition for convexity/concavity from below.

7.15 Differential and Linear Approximation Definition. The equation for the tangent to the graph of at the point is Slope of the tangent line T is exactly equal to the value of derivative

Differentials Increasing the value of the argument from to results in the value of the function changing from to . From the equation of the tangent line at , where is an error term. Definition. In case exists at point , the value is called function f’s differential at point . dx’s DO NOT cancel each other out! When dx is small, the function’s differential dy is a fairly good linear approximation to the actual change .