   Chapter 2, Problem 12CRQ

Chapter
Section
Textbook Problem

Fill in the blanks,12. a. Suppose f is continuous on [a, b] and f(a) < M < f(b). Then the Intermediate Value Theorem guarantees the existence of at least one number c in _________such that ________. b. If f is continuous on [a, b] and f(a)f(b) < 0, then there must be at least one solution of the equation _________ in the interval ________.

(a)

To determine

To fill: The statement “ f is continuous on [a,b] and f(a)<M<f(b) , then the Intermediate value theorem guarantees the existence of at least one number c in _________ such that __________”.

Explanation

If the function y=f(x) is continuous on the interval [a,b] where aandb are real numbers and a number M lies between f(a) and f(b) , then there must be at least one value c within [a,b] such that f(c)=M is called intermediate value theorem.

Here the function f must be continuous.

Assume a function is,

f(x)=x2 at [2,3] (1)

Substitute 2 for x in equation (1) to find the value of f(2) ,

f(2)=22=4((2)2=4)

(b)

To determine

To fill: The statement “If f is continuous on [a,b] and f(a)f(b)<0 , then there must be at least one solution of the equation _________ in the interval ____________”.

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