Suppose f(x) is continuous on (0,2) and never zero there, and f(1)<0 . What can you say about th

step 1 We are trying to show that this function has, when we take all x values between 0 and 2, produce

Suppose f(x) is continuous on (0,2) and never zero there,...

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Key Concepts

Continuity

Continuity ensures that there are no breaks or jumps in a function over an interval, meaning the function takes on every value between its minimum and maximum on that interval. This property is crucial in determining behavior such as sign consistency, as any change in sign would have to be accompanied by the function crossing zero.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous over an interval, then for any two values the function attains at the endpoints of the interval, it also attains every value in between. This theorem is essential in understanding that if a continuous function changes from negative to positive (or vice versa), it must pass through zero.

Sign Consistency on Intervals

Due to the continuity of the function and the Intermediate Value Theorem, a function that is never zero on an interval cannot change its sign. In this case, since the function is continuous, never zero, and is negative at a specific point, it must be negative throughout the entire interval. This concept highlights that a continuous function on a connected interval, which avoids zero, maintains a constant sign.

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