Which statement describes the Intermediate Value Theorem?

Which statement describes the Intermediate Value Theorem?

• A. If f is continuous on [a,b] and k lies between f(a) and f(b), there exists c in (a,b) such that f(c)=k.
• B. If f is continuous on [a,b], then f is differentiable on (a,b).
• C. If f is differentiable on (a,b), then f' takes all intermediate values.
• D. If f is continuous on [a,b], f(a)=f(b).

Answer: (A)

The main idea being tested is that a continuous function on a closed interval takes every value between its endpoint values. If you have a function continuous on [a, b], then for any horizontal level k that lies between f(a) and f(b), there must be some c in (a, b) with f(c) = k. This is the intuition: tracing the graph from a to b, you start at height f(a) and end at height f(b) without any jumps, so you can’t skip over any intermediate height. The statement in question captures this exact property, specifying a c between the endpoints where the function hits k. This explains why it’s the best description of the Intermediate Value Theorem. It directly asserts the guaranteed hitting of every intermediate value by a continuous function. Other statements describe different ideas. Continuity alone does not imply differentiability, so a claim about the derivative is a separate concept. The claim that the derivative takes all intermediate values is a result known as the Darboux property for derivatives, not the IVT. And simply having equal endpoint values does not articulate the IVT’s guarantee about hitting every intermediate value.

The main idea being tested is that a continuous function on a closed interval takes every value between its endpoint values. If you have a function continuous on [a, b], then for any horizontal level k that lies between f(a) and f(b), there must be some c in (a, b) with f(c) = k. This is the intuition: tracing the graph from a to b, you start at height f(a) and end at height f(b) without any jumps, so you can’t skip over any intermediate height. The statement in question captures this exact property, specifying a c between the endpoints where the function hits k.

This explains why it’s the best description of the Intermediate Value Theorem. It directly asserts the guaranteed hitting of every intermediate value by a continuous function.

Other statements describe different ideas. Continuity alone does not imply differentiability, so a claim about the derivative is a separate concept. The claim that the derivative takes all intermediate values is a result known as the Darboux property for derivatives, not the IVT. And simply having equal endpoint values does not articulate the IVT’s guarantee about hitting every intermediate value.