Important Theorems about continuous functions

1. Important Theorems about continuous functions • Extreme Value Theorem • Intermediate Value Theorem • Some applications

2. Intuitive Picture • Imagine graph of f a 2-dimensional profile of a mountain range • Tops of mountains correspond to relative maxima and bottoms of valleys to relative minima • Geologically these are high and low points of terrain in immediate vicinity • Just as geologist might be interested in finding highest mountain and deepest valley in entire mountain range, • Mathematician might be interested in finding the largest and smallest values of functions over entire domain

3. Definition of absolute extrema • Suppose that f is a function defined on a domain D containing c. Then • Absolute maximum value at c if f(c)  f(x) for all x  D • Absolute minimum value at c iff(c)  f(x) for all x  D

4. Extreme value theorem • Can find absolute extrema under certain hypotheses: • Extreme Value Theorem: If f is continuous on a closed interval [a,b], with - < a < b < , then f has an absolute maximum and an absolute minimum on [a,b]

5. Example No maximum or minimum value. However, on [-3,3], it hasboth.

6. Another Example

7. Conclusions about hypotheses • Conclude that hypothesis that interval be closed, [a,b], essential • Conclusion that f is continuous also essential:

8. Examples fulfilling hypotheses • f(x) = 2 - 3x where -5 < x < 8 • g(x) = sin(x) where 0 < x < 2p

9. Limitations of Extreme Value Theorem • Polynomial f(x)=x5 - 3x2 + 13 is continuous everywhere • Must have absolute max, min on [-1, 10] by theorem • Theorem doesn’t say where these occur • Extreme value theorem just an “existence theorem” • Learn tools for finding extrema later using the derivative

10. Intermediate Value Theorem: Intuition • Traveling on France’s TGV trains, you reach speed of 280 mi/hr. • How do you know at some point of train ride you were traveling 100 mi/hr? • To go from 0 to 280, must have passed through 100 mi/hr since speed of train changed continuously

11. Intermediate Value Theorem • Suppose that f is continuous on the closed interval [a,b]. If L is any real number between f(a) and f(b) then there must be at least one number c on the open interval (a,b) such that f(c) = L.

13. Limitations of IVT • The IVT is a powerful tool, but it has its limitations. To illustrate, suppose that d(t) represents the decibel level of Pork Chop's motorcycle engine, and suppose • d(0) = 100 and d(10) = 35, where t is measured in seconds. • d is a continuous function. • By IVT in the ten second interval between time t=0 and time t=35 Pork Chop's decibel level reached every value between 35 and 100. • It does NOT say anything about: • When or how many times (other than at least once) a particular decibel was attained. • Whether or not decibel levels bigger than 100 or less than 35 were reached.

14. The Difference Between VROOOOOOOOM and VROOOOOOOM. These graphs of PC's noise illustrate that very different behaviors are consistent with the hypothesis that d(t) is continuous and that its values at t=0 and t=10 are 100 and 35 respectively.

15. Important Special Case • If f is continuous on [a,b] and if f(a) and f(b) have opposite signs then there must be an x in [a,b] so that f(x)=0 • Example: Show that there is an x in [1,2] so that x3-4x+1=0 • Example: Show that polynomials of odd degree always have a root. That is if P is of odd degree then there must be at least on x so that P(x)=0

16. Other Examples • Use the IVT to show that the following equations have solutions