Question

Suppose that the graph of a differentiable function $f(x)$ has a horizontal tangent at $x=a$ . Can anything be said about the linearization of $f$ at $x=a ?$ Give reasons for your answer.

Name: Problem 64, Chapter 3: Thomas Calculus
Uploaded: 2020-09-15T20:12:45-08:00
Duration: 2 min 35 s
Channel: Ahmad Reda
Description: Suppose that the graph of a differentiable function f(x) has a horizontal tangent at x=a . Can anything be said about the linearization of f at x=a ? Give reasons for your answer.

   Suppose that the graph of a differentiable function $f(x)$ has a horizontal tangent at $x=a$ . Can anything be said about the linearization of $f$ at $x=a ?$ Give reasons for your answer.

Thomas Calculus

George B. Thomas,… 11th Edition

Chapter 3, Problem 64

Instant Answer

Solved by Expert Ahmad Reda

09/15/2020

Step 1

This means that the derivative of the function at $x=a$ is zero, i.e., $f'(a)=0$. Show more…

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Suppose that the graph of a differentiable function $f(x)$ has a horizontal tangent at $x=a$ . Can anything be said about the linearization of $f$ at $x=a ?$ Give reasons for your answer.

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Key Concepts

Differentiability

Differentiability implies that the function has a well-defined derivative at the point in question, meaning it can be locally approximated by its tangent line. This is a prerequisite for defining and using linearization, which relies on the smooth behavior of the function near the given point.

Horizontal Tangent

A horizontal tangent at a given point means that the derivative of the function at that point is zero. This indicates that the slope of the tangent line is zero, which typically occurs at local extrema or saddle points.

Linearization

Linearization approximates a function at a point using the tangent line, expressed as L(x) = f(a) + f'(a)(x - a). When the tangent is horizontal, f'(a) is zero, resulting in a constant linearization L(x) = f(a). This shows that near the point, the function is locally approximated by a constant function equal to its value at that point.

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