Question

is it true that a continuous function that is never zero on an interval never changes sign on that interval? give reasons for your answer. choose the correct answer below. yes. if the function is defined piece - wise, it may change signs without equaling zero. yes. by the intermediate value theorem, if a function changed sign, its value was zero at some point. no. if the function is continuous, it cannot change sign. no. by the intermediate value theorem, a function could change sign without equaling zero.

Explanation:

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 \(f(c)=k\). If a continuous function changes sign on an interval (e.g., \(f(x)\) is positive at one end - point and negative at the other end - point of the interval), then by the Intermediate - Value Theorem, there must be a point in the interval where \(f(x) = 0\) since \(0\) is between a positive and a negative number. So, a continuous function that is never zero on an interval never changes sign on that interval.

Answer:

Yes. By the Intermediate Value Theorem, if a function changed sign, its value was zero at some point.