12. classify the following statement as true or false. explain.\nif ( f(x) ) is continuous everywhere and (…

12. classify the following statement as true or false. explain.\nif ( f(x) ) is continuous everywhere and ( f(a), f(b)>0 ), then there is no root of ( f(x) ) in the interval\n(a, b).

12. classify the following statement as true or false. explain.\nif ( f(x) ) is continuous everywhere and ( f(a), f(b)>0 ), then there is no root of ( f(x) ) in the interval\n(a, b).

Answer

Explanation:

Step1: Recall Intermediate Value Theorem

The Intermediate Value Theorem states that if (y = f(x)) is continuous on a closed interval ([a,b]), and (k) is a number between (f(a)) and (f(b)), then there exists at least one number (c\in(a,b)) such that (f(c)=k). A root of (f(x)) is a value (x = c) where (f(c)=0).

Step2: Consider a counter - example

Let (f(x)=(x - \frac{a + b}{2})^2+1). This function is continuous everywhere (since it is a polynomial). Let (a = 0) and (b = 2). Then (f(0)=(0 - 1)^2+1=2>0) and (f(2)=(2 - 1)^2+1=2>0). But if we consider (f(x)=x(x - 2)+1=x^{2}-2x + 1=(x - 1)^2), (f(0)=1>0), (f(2)=1>0), and (f(1)=0) (a root in the interval ([0,2])).

Answer:

False.