Question

suppose that $f(x)$ is a continuous function on the interval $2,6$ with $f(2)=-1$ and $f(6)=1$. determine which choice best describes the following statement. \$f(x)=0$ for some $x$ in the interval $2,6$\

○ always false
○ sometimes true and sometimes false
○ always true

Explanation:

Step1: Apply 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

Here, \(a = 2\), \(b = 6\), \(f(2)=- 1\), \(f(6)=1\), and \(k = 0\). Since \(0\) is between \(-1\) and \(1\) and \(f(x)\) is continuous on \([2,6]\).

Answer:

Always true