Question

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

Answer: The graph of the linear approximation at a local maximum is a horizontal line that touches the function's graph at the point of the local maximum, representing the function's behavior near that point.

Step-by-step solution

Understand linear approximation

Linear approximation is a technique used to approximate the value of a function near a certain point, using the tangent line at that point. In simpler terms, we "approximate" the function with a straight line that closely resembles the function's behavior around a specific point.

Recall the properties of a local maximum

A local maximum of a function is a point in which the function's value is greater than or equal to the values of the function around that point within a certain interval. In other words, it is a high point or "peak" in the function's graph. For a differentiable function at a local maximum, the first derivative of the function (the slope of the tangent line) is 0.

Describe the graph of the linear approximation

Since the first derivative is 0 at the local maximum, this means that the tangent line (linear approximation) at that point will be horizontal. To describe the graph of the linear approximation, we can say that it is a horizontal line that touches the graph of the function at the point of the local maximum. This line accurately represents the behavior of the function near that point.