- Give an example of a continuous function defined on which does not achieve its maximum on.
- Give an example of a continuous function defined on which is bounded but which does not achieve either its maximum or its minimum.
- Give an example of a discontinuous function defined on which is bounded but does not achieve either its maximum or its minimum.
- Give an example of a continuous function defined on [0,1) ∪ (1,2] which is positive at 2, negative at 0 but is not equal to zero for any value of x.
- Let f= x
^{5}+ ax^{4}+ bx^{3}+ cx^{2}+ dx + e where a,b,c,d, and e are numbers. Show there exists real x such that f= 0 . - Give an example of a function which is one to one but neither strictly increasing nor strictly decreasing.
- Show that the function f= x
^{n}− a, where n is a positive integer and a is a number, is continuous. - Use the intermediate value theorem on the function f= x
^{7}− 8 to showmust exist. State and prove a general theorem about n^{th}roots of positive numbers. - Prove is irrational. Hint: Suppose= p∕q where p,q are positive integers and the fraction is in lowest terms. Then 2q
^{2}= p^{2}and so p^{2}is even. Explain why p = 2r so p must be even. Next argue q must be even. - Let f= x −for x ∈ ℚ, the rational numbers. Show that even though f< 0 and f> 0, there is no point in ℚ where f= 0 . Does this contradict the intermediate value theorem? Explain.
- A circular hula hoop lies partly in the shade and partly in the hot sun. Show
there exist two points on the hula hoop which are at opposite sides of the
hoop which have the same temperature. Hint: Imagine this is a circle and
points are located by specifying their angle, θ from a fixed diameter. Then
letting Tbe the temperature in the hoop, T= T. You need to have T= Tfor some θ. Assume T is a continuous function of θ.
- A car starts off on a long trip with a full tank of gas. The driver intends to drive the car till it runs out of gas. Show that at some time the number of miles the car has gone exactly equals the number of gallons of gas in the tank.
- Suppose f is a continuous function defined on which mapsinto. Show there exists x ∈such that x = f. Hint: Consider h≡ x−fand the intermediate value theorem. This is a one dimensional version of the Brouwer fixed point theorem.
- Let f be a continuous function on such that f= f. Let n be a positive integer larger than 2. Show there must exist c ∈such that f= f. Hint: Consider h≡ f− f. Consider the subintervalsfor k = 1,,n−1. You want to show that h equals zero on one of these intervals. If h changes sign between two successive intervals, then you are done. Assume then, that this does not happen. Say h remains positive. Argue that f< f. Thus f> f= f. It follows that h< 0 but h> 0.

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