Maximum and Minimum Values Section 4 1 Alex

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Maximum and Minimum Values (Section 4. 1) Alex Karassev

Maximum and Minimum Values (Section 4. 1) Alex Karassev

Absolute maximum values l A function f has an absolute maximum value on a

Absolute maximum values l 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) S c x

Absolute minimum values l A function f has an absolute minimum value on a

Absolute minimum values l 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) S c x

Example: f(x) = x 2 l l l S = (-∞, ∞) No absolute

Example: f(x) = x 2 l l l S = (-∞, ∞) No absolute maximum Absolute minimum: f(0) = 0 at c = 0 y x 0

Example: f(x) = x 2 l l l S = [0, 1] Absolute maximum

Example: f(x) = x 2 l l l S = [0, 1] Absolute maximum f(1) = 1 at c = 1 Absolute minimum: f(0) = 0 at c = 0 y x 0 1

Example: f(x) = x 2 l l l S = (0, 1] Absolute maximum

Example: f(x) = x 2 l l l S = (0, 1] Absolute maximum f(1) = 1 at c = 1 No absolute minimum, although function is bounded from below: 0 < x 2 for all x in (0, 1] ! y x 0 1

Local maximum values l A function f has a local maximum value at a

Local maximum values l 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 l A function f has a local minimum value at a

Local minimum values l 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

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 -1

Applications l l Curve sketching Optimization problems (with constraints), for example: q q Finding

Applications l l Curve sketching Optimization problems (with constraints), for example: q q 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

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

Extreme Value Theorem - Examples y y y = f(x) x a c MAX

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

Continuity is important y x -1 0 No absolute maximum or minimum on [-1,

Continuity is important y x -1 0 No absolute maximum or minimum on [-1, 1] 1

Closed interval is important l l f(x) = x 2, S = (0, 1]

Closed interval is important l l f(x) = x 2, S = (0, 1] No absolute minimum in (0, 1] y x 0 1

How to find max and min values? l Absolute maximum or minimum values of

How to find max and min values? l 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 l 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)

Fermat's Theorem y horizontal tangent line at the point of local max (or min) y = f(x) x c l 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! l If f ′(c) = 0 it

Converse of Fermat's theorem does not hold! l If f ′(c) = 0 it does not mean that c is a point of local maximum or minimum l Example: f(x) = x 3, f ′(0) = 0, but 0 is not a point of local max or min l Nevertheless, points c where f ′(c) = 0 are "suspicious" points (for local max or min) y x

Problem: f′ not always exists l f(x) = |x| l It has local (and

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

Critical numbers l Two kinds of "suspicious" points (for local max or min): f′(c)

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

Critical numbers – definition l A number c is called a critical number of

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

Closed Interval Method l The method to find absolute maximum or minimum of a

Closed Interval Method l The method to find absolute maximum or minimum of a continuous function, defined on a closed interval [a, b] l Based on the fact that absolute maximum or minimum q 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) q or is attained at one of the endpoints

Closed Interval Method l To find absolute maximum and minimum of a function f,

Closed Interval Method l To find absolute maximum and minimum of a function f, continuous on [a, b]: q q Find critical numbers inside (a, b) l Find derivative f′ (x) l Solve equation f′ (x)=0 for x and choose solutions which are inside (a, b) l Find numbers in (a, b) where f′ (x) d. n. e. Suppose that c 1, c 2, …, ck are all critical numbers in (a, b) l The largest of f(a), f(c 1), f(c 2), …, f(ck), f(b) is the absolute maximum of f on [a, b] l The smallest of these numbers is the absolute minimum of f on [a, b]