True or False? In Exercises 65-68 , determine whether the...

True or False? In Exercises $65-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $x=c$ is a critical number of the function $f,$ then it is also a critical number of the function $g(x)=f(x-k),$ where $k$ is a constant.

True or False? In Exercises $65-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $x=c$ is a critical number of the function $f,$ then it is also a critical number of the function $g(x)=f(x-k),$ where $k$ is a constant.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Critical Numbers

Critical numbers are values in the domain of a function where its derivative is either zero or undefined. These points are where the function may have local extrema or inflection points, and they are crucial in the study of function behavior and optimization.

Function Transformations (Horizontal Shifts)

A horizontal shift of a function, given by an expression like g(x) = f(x - k), moves the graph of f horizontally by k units. This transformation affects the location of the critical numbers by shifting them accordingly; specifically, a critical number of f at x = c would correspond to a critical number of g at x = c + k, not at x = c.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. In the context of a horizontally shifted function such as g(x) = f(x - k), the derivative is found by multiplying the derivative of f evaluated at x - k by the derivative of (x - k), which is 1. Thus, the zeros of g's derivative occur when f's derivative at the shifted point is zero.

Transcript

Please give Ace some feedback