Question

question 8
suppose 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.
( f(x)=0 ) for some ( x ) in the interval (-3,1)
always false
sometimes true and sometimes false
always true
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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