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A relative maximum is a point on a function where the value of the function is higher than all other nearby points. Imagine hiking up a hill. The top of the hill is a relative maximum because it's the highest point you can reach before you start going downhill again. At a relative maximum, the function stops increasing and starts decreasing.

If a smooth curve has two relative maximum points, the function will first increase to the top of the first hill, then decrease before starting to go uphill again toward the second relative maximum. The area between the two hills will have a dip, where the function switches from decreasing to increasing. This dip is known as a relative minimum.

A relative minimum is the opposite of a relative maximum. It’s where the function value is lower than all the nearby points. Think of it as a valley between hills on your hiking path. At this point, the function stops decreasing and starts increasing.

If you have two hills (relative maximums), there must be a valley (relative minimum) between them. So for the curve to have two relative maximums, it will always have at least one relative minimum in between, ensuring there's a point where the function transitions from decreasing to increasing.

An inflection point is where the concavity of the function changes. Concavity describes whether the curve bends upwards (concave up) or downwards (concave down). At an inflection point, the function switches from bending one way to bending the other.

Between any two relative extreme points—whether they are maximums, minimums, or one of each—the concavity must change. Consequently, there will be an inflection point between these extremes. Identifying an inflection point helps you understand where the curve's shape shifts.

A smooth curve is a function that has no sharp corners or breaks. Everything blends smoothly. In calculus, this means the function and its derivatives are continuous. You can draw it without lifting your pencil from the paper.

For our problem, working with a smooth curve guarantees all transitions from increasing to decreasing or changes in concavity happen gradually, allowing the identification of relative maximums, minimums, and inflection points more effectively.

Concavity describes how a curve bends. If the curve bends upwards like a bowl, it’s called concave up. If it bends downwards like a dome, it’s called concave down. The concavity of a function gives insight into its shape between extreme points.

When analyzing a smooth curve, concavity helps to distinguish inflection points and better understand the overall form of the function. Changes in concavity are key to pinpointing where the curve potentially has inflection points, particularly between relative maximum and minimum points.