suppose (f) is continuous on ((-infty,infty)) and the following data is known: (f(-1)=2,f(0)=4,f(1)=…

suppose (f) is continuous on ((-infty,infty)) and the following data is known: (f(-1)=2,f(0)=4,f(1)= - 1,f(2)=2,f(3)=-6). which of the following statements is true? (f(x) = 0) has exactly three solutions. (f(x)=0) has at most two solutions. (f(x)=0) has at least three solutions. (f(x)=0) has at most three solutions. (f(x)=0) has at exactly two solutions.

suppose (f) is continuous on ((-infty,infty)) and the following data is known: (f(-1)=2,f(0)=4,f(1)= - 1,f(2)=2,f(3)=-6). which of the following statements is true? (f(x) = 0) has exactly three solutions. (f(x)=0) has at most two solutions. (f(x)=0) has at least three solutions. (f(x)=0) has at most three solutions. (f(x)=0) has at exactly two solutions.

Answer

Explanation:

Step1: Apply Intermediate - Value Theorem

The Intermediate - Value Theorem states that if a function (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 the interval ((a,b)) such that (f(c)=k). We know (f(-1) = 2) and (f(0)=4), (f(1)= - 1), (f(2)=2), (f(3)=-6). Since (f(0) = 4) and (f(1)=-1), and (0) is between (4) and (- 1), by the Intermediate - Value Theorem, there exists a number (c_1\in(0,1)) such that (f(c_1)=0). Since (f(1)=-1) and (f(2) = 2), and (0) is between (-1) and (2), by the Intermediate - Value Theorem, there exists a number (c_2\in(1,2)) such that (f(c_2)=0). Since (f(2)=2) and (f(3)=-6), and (0) is between (2) and (-6), by the Intermediate - Value Theorem, there exists a number (c_3\in(2,3)) such that (f(c_3)=0).

Answer:

(f(x) = 0) has at least three solutions.