question 8\nsuppose that ( f(x) ) is a continuous function on the interval (-3,1) with ( f(-3)= - 7) and (…

question 8\nsuppose that ( f(x) ) is a continuous function on the interval (-3,1) with ( f(-3)= - 7) and ( f(1)=7). determine which choice best describes the following statement.\n( f(x)=0 ) for some ( x ) in the interval (-3,1)\nalways false\nsometimes true and sometimes false\nalways true\nquestion help: video message instructor\nsubmit question

question 8\nsuppose that ( f(x) ) is a continuous function on the interval (-3,1) with ( f(-3)= - 7) and ( f(1)=7). determine which choice best describes the following statement.\n( f(x)=0 ) for some ( x ) in the interval (-3,1)\nalways false\nsometimes true and sometimes false\nalways true\nquestion help: video message instructor\nsubmit question

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 the interval ((a,b)) such that (f(c)=k).

Step2: Identify values of (a), (b), (f(a)) and (f(b))

Here, (a=-3), (b = 1), (f(-3)=-7) and (f(1)=7). The number (k = 0) is between (f(-3)=-7) and (f(1)=7).

Step3: Apply the theorem

Since (f(x)) is continuous on ([-3,1]) and (0) is between (f(-3)=-7) and (f(1)=7), there exists at least one (x) in the interval ([-3,1]) such that (f(x)=0).

Answer:

Always true