Behavior of a Function and Its Derivative
Have you ever noticed how the graph of a function can move up, down, or even flatten out for a moment? This behavior, called the monotonicity of a function, is closely related to its first derivative.
Imagine you are walking along the curve of a graph from left to right.
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When you are climbing, it means the function is increasing.
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When you are descending into a valley, it means the function is decreasing.
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When you are at the top of a hill or the bottom of a valley, you are at a stationary point.
Geometrically, the first derivative, , is the gradient of the tangent line to the curve at that point. So, we can determine the function's behavior by looking at the sign of its gradient.