What can be inferred about the graph of a function if it has a local maximum?

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Question: What can be inferred about the graph of a function if it has a local maximum?

Options:

Correct Answer: The derivative at that point is zero.

Solution:

At a local maximum, the derivative of the function is zero, indicating a horizontal tangent.

What can be inferred about the graph of a function if it has a local maximum?

What can be inferred about the graph of a function if it has a local maximum?

Step 1: Understand what a local maximum is. A local maximum is a point on the graph where the function reaches a peak compared to nearby points.
Step 2: Recognize that at a peak, the slope of the graph changes direction. This means the graph goes up before the peak and down after the peak.
Step 3: The slope of the graph is represented by the derivative of the function. When the slope is zero, it means the graph is flat (horizontal) at that point.
Step 4: Therefore, at a local maximum, the derivative of the function is equal to zero, indicating a horizontal tangent line.

Local Maximum – A point on the graph where the function value is higher than the values of the function at nearby points.
Derivative – The slope of the tangent line to the graph of the function, which indicates the rate of change of the function.
Horizontal Tangent – A tangent line that is parallel to the x-axis, indicating that the derivative at that point is zero.