A function is increasing on A if f(x)<=f(y) for all x <y in A. Show that the intermediate value theorem does have a converse if we assume f is increasing on [a,b].

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We'll prove this by contradiction.
Suppose that f is not continuous and let a<c<b.
Let e>0 .Since f is ...

Solution Summary

The Intermediate Value Theorem is proven not to have a converse over a given interval. The solution is concise.

Let f : [0,1]-->[0,1] be a continuous function. Show that there exists a real number x in [0,1] such that f(x)=x (apply the intermediatevaluetheorem to the function f(x) -x). This point x is known as a fixed point of f, and this result is a basic example of a fixed point theorem.

Suppose that functions f,g : [a,b] -> R are continuous, satisfy f(a) <= g(a)
and f(b) >= g(b). Then there exists a real number c in [a,b] such that f(c) = g(c).
Label the statement as true or false. If it is true, prove it. If not, give an example of why it is false and if possible, correct it to make it true.

1. Using the intermediateValuetheorem, show that the function f has a zero between a and b.
f(x)=x^3+3x^2-9x-13, a=-5, b=-4
2. For the function h(x)=5x/((x+6)(x-4)), solve the following equation
h(x)=0

For numbers a1,....,an, define
p(x) = a1x +a2x^2+....+anx^n for all x. Suppose that:
(a1)/2 + (a2)/3 +....+ (an)/(n+1) = 0
Prove that there is some point x in the interval (0,1) such that p(x) = 0

21) Use synthetic division to find: f(?2)
f(x)=2x3?3x2+7x?12 21)
22) For the function f(x) = ?x3 + 8x2 ? 40
Show how to use synthetic division to find out if -1 is a zero.
22)
23) One of the zeros of the function f(x) = ? 7x2 + 17x ?15 is 2-i. Find all the other zeros.
23) ____________________
24) Use the intermediate val

1. Use the intermediatevaluetheorem to show that the polynomial function has a zero in the given interval.
F(x)=x^5-x^4+7x^3-8x^2-16x+13; [1.3,1.7]
Find value of f (1.3) ____ (simplify)
Find value of f (1.7) ______ (simplify)
2. Information is given about the polynomial f(x) whose coefficients a

At 8am on Saturday, a man begins running up the side of a mountain to his weekend campsite. On Sunday at 8am, he runs back down the mountain. It takes him 20 minutes to run up, but only 10 minutes to run down. At some point on his way down, he realizes that he passed the same exact place at exactly the same time on Saturday.

Give an example to show that the intermediatevaluetheorem becomes false if the hypothesis that f is continuous is replaced with f being montone, or with f being strictly monotone. (can use the same counterexample for both)

Let h be a differentiable function defined on the interval [0,3], and assume that h(0)=1 h(1)=2 and h(3)=2.
a- argue that there exists a point d belong to [0,3] where h(d)=d.
b-argue that at some point c we have h'(c)=1/3.
c-argue that h'(x)=1/4 at some point in the domain.