When is there no critical points?


When is there no critical points?

Critical Point: Definition

Consider a function {eq}f(x) {/eq}. Suppose that the {eq}n^{th} {/eq} derivative of the function is found in a form {eq}f^{(n)}(x) {/eq}. A critical point is a point at which:

  • The derivative is equal to 0, that is, {eq}f^{(n)}(x) = 0 {/eq}
  • Or the derivative doesn't exist, {eq}f^{(n)}(x)~\nexists {/eq}

Answer and Explanation:

Let's take, for example, a function {eq}f(x) {/eq} and its simplest derivative, the first derivative:

{eq}f'(x) {/eq}

a. There will be no critical points if no {eq}x {/eq} value satisfies {eq}f(x) = 0 {/eq}

b. Or our derivative function is defined {eq}\forall~x {/eq} values.

Therefore, there are no critical points if {eq}\boxed{f^{(n)}\neq 0~\forall~x} {/eq} or {eq}\boxed{f^{(n)} \exists~\forall~x} {/eq}

Learn more about this topic:

Finding Critical Points in Calculus: Function & Graph

from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9

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