Maximum and Minimum Values Section 3 1 Alex
- Slides: 24
Maximum and Minimum Values (Section 3. 1) Alex Karassev
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 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 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 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 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 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 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 -1
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 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 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, 1] 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 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) 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 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 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) = 0 q f′(c) does not exists q
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 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, 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]
Example l Find the absolute maximum and minimum values of f(x) = x/(x 2+1) on the interval [0, 2]
Solution Find the absolute maximum and minimum values of f(x) = x/(x 2+1) on the interval [0, 2] l Find f′(x): l Critical numbers: f′(x) = 0 ⇔ 1 – x 2 = 0 l So x = 1 or x = – 1 l However, only 1 is inside [0, 2] l Now we need to compare f(0), f(1), and f(2): l f(0) = 0, f(1) = 1/2, f(2)= 2/5 l Therefore 0 is absolute minimum and 1/2 is absolute maximum
- Local maximum
- Quadratic function maximum
- Quadratic model
- Maximum and minimum values of quadratic functions
- Minimum mode of 8086
- Maxmin
- Relative maximum and minimum
- Maximum and minimum derivative
- What is the difference between maximum and minimum
- Equation of a quadratic function
- Maximum clearance formula
- Mml of hole and shaft
- Minimum enthalpy maximum entropy
- Bellani-six thermometer.
- Partial order
- Rumus nilai maksimum dan minimum
- Local maximum
- Maximum value
- Maximum parsimony vs maximum likelihood
- Maximum likelihood vs maximum parsimony
- Minimum values
- Instumental values
- An individual's enduring tendency to feel
- Western vs eastern values
- The format character %8d will contain ____ decimals.