1. Let f be a continuous function dened on the closed interval [a; b], as in the
ID: 2866987 • Letter: 1
Question
1. Let f be a continuous function dened on the closed interval [a; b], as in the statement of the IVT. Use
the IVT to show that the range of f contains a closed interval.
2. Use the IVT to prove the existence of sqrt 2. (Hint: what does it mean to say that a number s is a square
root of 2?)
3. Does the theorem remain true if we drop the assumption that f is continuous? Why or why not?
4. Does the conclusion of the theorem still hold if L is not required to lie between f(a) and f(b)? Why
or why not?
5. Suppose instead that we assume f is a continuous function on the open interval (a; b). Can we still
formulate a version of the IVT which is always true? Why or why not?
Explanation / Answer
2)f(x) = x^2 is continuous, so we can apply the IVT.
f(1) = 1^2 = 1
f(2) = 2^2 = 4
By the IVT, there is a c between 1 and 2 such that f(c) = 2. That is, c^2 = 2, so c = sqrt(2).
1)a closed interval [a,b] such that f(a)<0 and f(b)>0. If I let a=0, then f(a)=?s<0 (since s>0),hence it follows
3)if we drop in this theorem the local boundedness. assumption on S the theorem remains true if we replace the term equivalent local .... Let the stochastic base (?, F, P) be such that there are two independent stopping ... we have exhibited a sigma-martingale for which there does not exist hence it is true
4)Let L be the set of all x in [a, b] such that f(x) < u. Then L is non-empty since a is an element of L, and L is bounded above by b. Hence, by completeness, the supremum c = sup L exists. That is, c is the lowest number that is greater than or equal to every member of L
5)Consider a real-valued, continuous function f on a closed interval [a,b] with f(a) = f(b). ... If f is convex or concave, then the right- and left-hand derivatives exist at every inner value from a to b is possible , hence it is true
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.