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

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

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

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Key Concepts

Sign Preservation in Continuous Functions

For a continuous function that never equals zero on an interval, the property of sign preservation follows: if the function is positive at one point, it must remain positive throughout the entire interval, as any change would force the function to pass through zero. This concept is an application of the Intermediate Value Theorem and ensures the function maintains a consistent sign.

Continuity

Continuity of a function means that its graph can be drawn without lifting the pen from the paper. This property ensures that the function's values change gradually without any abrupt jumps, which is crucial in analyzing how the sign (positive or negative) behaves over an interval.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a given interval, then it takes every value between its values at the endpoints of the interval. In the context of sign changes, this theorem implies that if a continuous function were to change from positive to negative (or vice versa), it would have to cross zero, which is not possible here since the function is never zero.

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