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Chapter 4: Problem 2

Suppose you find the linear approximation to a differentiable function at a local maximum of that function. Describe the graph of the linear approximation.

Short Answer

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Question: Describe the graph of a linear approximation to a differentiable function at a local maximum. Answer: The graph of the linear approximation at the local maximum of a differentiable function is a horizontal line passing through the point of local maximum.

Step by step solution

01

Understanding local maximum and differentiable functions

A local maximum of a differentiable function is a point where the function achieves its highest value in some neighborhood (an interval) around that point. A differentiable function is a function that has a derivative at every point in its domain. The derivative represents the slope of the tangent line to the function at a specific point.
02

Finding the tangent line at the local maximum

To find the tangent line at the local maximum, we need to compute the derivative of the function and evaluate it at the local maximum point. Since the local maximum is the highest point in a neighborhood, the tangent line at this point will be horizontal with a slope of 0.
03

Describing the graph of the linear approximation

The linear approximation at the local maximum is based on the tangent line equation at that point. Since the tangent line at the local maximum is horizontal with a slope of 0, the graph of the linear approximation will be a horizontal line that intersects the function exactly at the local maximum point. Thus, the graph of the linear approximation at the local maximum of a differentiable function is a horizontal line passing through the point of local maximum.

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