   Chapter 2, Problem 24RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The equation x10 – 10x2 + 5 = 0 has a root in the interval (0, 2).

To determine

Whether the statement, “the equation x1010x2+5=0 has a root in the interval (0, 2)” is true or false.

Explanation

Theorem used: The Intermediate value Theorem

Suppose that if f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a)f(b). Then there exists a number c in (a, b) such that f(c)=N.

Proof:

To show there is a root of the equation x1010x2+5=0 in the interval (0, 2), it is enough to show that there is a number c between 0 and 2 for which f(c)=0.

Consider the function f(x)=x1010x2+5.

Here, f(x) is a polynomial function and it is continuous everywhere on the interval (0,2) and take a=0, b=2 and N=0.

Substitute 0 for x in f(x),

f(0)=(0)1010(0)2+5=00+5=5

This implies that, f(0)=5>0.

Substitute 2 for x in f(x),

f(2)=(2)1010(2)2+5=102410(4)+5=102440+5=989

This implies that, f(2)=989>0

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