Maximum and Minimum Values (Section 3.1) Alex Karassev

Absolute maximum values A function f has an absolute maximum value on a set S at a point c in S if f(c) ≥ f(x) for all x in S y y = f(x) f(c) x S c

Absolute minimum values A function f has an absolute minimum value on a set S at a point c in S if f(c) ≤ f(x) for all x in S y y = f(x) f(c) x S c

Example: f(x) = x2 S = (-∞, ∞) No absolute maximum Absolute minimum: f(0) = 0 at c = 0 y x

Example: f(x) = x2 S = [0,1] Absolute maximum f(1) = 1 at c = 1 Absolute minimum: f(0) = 0 at c = 0 y x 1

Example: f(x) = x2 S = (0,1] Absolute maximum f(1) = 1 at c = 1 No absolute minimum, although function is bounded from below: 0 < x2 for all x in (0,1] ! y x 1

Local maximum values A function f has a local maximum value at a point c if f(c) ≥ f(x) for all x near c (i.e. for all x in some open interval containing c) y y = f(x) x c

Local minimum values A function f has a local minimum value at a point c if f(c) ≤ f(x) for all x near c (i.e. for all x in some open interval containing c) y y = f(x) x c

Example: y = sin x f(x) = sin x has local (and absolute) maximum at all points of the form π/2 + 2πk, and local (and absolute) minimum at all points of the form -π/2 + 2πk, where k is an integer 1 - π/2 π/2 -1

Applications Curve sketching Optimization problems (with constraints), for example: Finding parameters to minimize manufacturing costs Investing to maximize profit (constraint: amount of money to invest is limited) Finding route to minimize the distance Finding dimensions of containers to maximize volumes (constraint: amount of material to be used is limited)

Extreme Value Theorem If f is continuous on a closed interval [a,b], then f attains absolute maximum value f(cMAX) and absolute minimum value f(cMIN) at some numbers cMAX and cMIN in [a,b]

Extreme Value Theorem - Examples y y y = f(x) y = f(x) x x a cMAX cMIN b a cMIN cMAX= b Both absolute max and absolute min are attained in the open interval (a,b) at the points of local max and min Absolute maximum is attained at the right end point: cMAX = b

Continuity is important x -1 1 No absolute maximum or minimum on [-1,1]

Closed interval is important f(x) = x2, S = (0,1] No absolute minimum in (0,1] y x 1

How to find max and min values? Absolute maximum or minimum values of a function, continuous on a closed interval are attained either at the points which are simultaneously the points of local maximum or minimum, or at the endpoints Thus, we need to know how to find points of local maximums and minimums

Fermat's Theorem y horizontal tangent line at the point of local max (or min) y = f(x) x c If f has a local maximum or minimum at c and f′(c) exists, then f′(c) = 0

Converse of Fermat's theorem does not hold! If f ′(c) = 0 it does not mean that c is a point of local maximum or minimum Example: f(x) = x3, f ′(0) = 0, but 0 is not a point of local max or min Nevertheless, points c where f ′(c) = 0 are "suspicious" points (for local max or min) y x

Problem: f′ not always exists f(x) = |x| It has local (and absolute) minimum at 0 However, f′ (0) does not exists! y x

Critical numbers Two kinds of "suspicious" points (for local max or min): f′(c) = 0 f′(c) does not exists

Critical numbers – definition A number c is called a critical number of function f if the following conditions are satisfied: c is in the domain of f f′(c) = 0 or f′(c) does not exist

Closed Interval Method The method to find absolute maximum or minimum of a continuous function, defined on a closed interval [a,b] Based on the fact that absolute maximum or minimum either is attained at some point inside the open interval (a,b) (then this point is also a point of local maximum or minimum and hence is a critical number) or is attained at one of the endpoints

Closed Interval Method To find absolute maximum and minimum of a function f, continuous on [a,b]: Find critical numbers inside (a,b) Find derivative f′ (x) Solve equation f′ (x)=0 for x and choose solutions which are inside (a,b) Find numbers in (a,b) where f′ (x) d.n.e. Suppose that c1, c2, …, ck are all critical numbers in (a,b) The largest of f(a), f(c1), f(c2), …, f(ck), f(b) is the absolute maximum of f on [a,b] The smallest of these numbers is the absolute minimum of f on [a,b]

Example Find the absolute maximum and minimum values of f(x) = x/(x2+1) on the interval [0,2]

Solution Find f′(x): Critical numbers: f′(x) = 0 ⇔ 1 – x2 = 0 Find the absolute maximum and minimum values of f(x) = x/(x2+1) on the interval [0,2] Solution Find f′(x): Critical numbers: f′(x) = 0 ⇔ 1 – x2 = 0 So x = 1 or x = – 1 However, only 1 is inside [0,2] Now we need to compare f(0), f(1), and f(2): f(0) = 0, f(1) = 1/2, f(2)= 2/5 Therefore 0 is absolute minimum and 1/2 is absolute maximum