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.

coose the correct answer below.

○ yes. if the function is defined piecewise, it may change signs without equaling zero.
○ no. by the intermediate value theorem, a function could change sign 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.

Explanation:

The Intermediate Value Theorem states that if a function \( f \) 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 \([a, b]\) such that \( f(c)=k \). For a continuous function on an interval, if it changed sign (e.g., \( f(a)>0 \) and \( f(b)<0 \) or vice - versa), then by the Intermediate Value Theorem, there would be some point \( c \) in the interval where \( f(c) = 0 \) (since 0 is between \( f(a) \) and \( f(b) \) in this case). So if a continuous function is never zero on an interval, it can't change sign. The first option is wrong because a piece - wise continuous function still follows the Intermediate Value Theorem. The second option misapplies the Intermediate Value Theorem. The fourth option is wrong as the continuity alone doesn't prevent sign change unless we consider the non - zero condition along with the Intermediate Value Theorem. The third option correctly applies the Intermediate Value Theorem: if a continuous function changed sign, it would have to be zero at some point (by IVT), so if it's never zero, it never changes sign.

Answer:

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