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17Calculus Derivatives - Critical Points

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Critical Numbers and Critical Points

In this section, we discuss critical numbers and critical points, what they are and how to find them.

Topics You Need To Understand For This Page

derivatives graphing 1st derivative

Critical numbers are x-values of functions where something special happens. These x-values must be in the domain. Two possible things can occur at critical number, \(x=c\).
1. If the derivative exists at \(x=c\), then the derivative at \(x=c\) is zero.
2. The derivative does not exist at \(x=c\).
If one of those two things happen, then \(x=c\) is called a critical value or critical number.
Note: If the term 'critical point' is used, then that refers to the point \((c,f(c))\), not just the number \(x=c\).

To find the critical numbers, we take the derivative, set it equal to zero and solve of the x-values. If those x-values are in the domain of the function, then they are critical values. Finally, we look at the derivative and determine where the derivative doesn't exist. If those points are in the domain, then those points are also considered critical points.

Important Thing to Remember

The value \(x=c\) must be in the domain of the function. If it is not in the domain, it cannot be a critical number.

The interesting thing that happens at critical points is that the function levels out at these points giving us three possible situations.
1. maximum
2. minimum
3. saddlepoint

Maximums and minimums, which may be either relative or absolute, are points where the graph is either increasing or decreasing, it levels off and changes direction.
Saddlepoints occur where a graph is either increasing or decreasing, it levels off at a point then the graph continues on in the same direction. The graph shows these types of points.

Okay, let's work some practice problems.

Math Word Problems Demystified

Practice

Unless otherwise instructed, determine the critical numbers of these functions.

\( f(x)=x^{1/3}-x^{-2/3} \)

Problem Statement

Determine the critical numbers for the function \( f(x)=x^{1/3}-x^{-2/3} \).

Final Answer

\(x=0\) and \(x=-2\)

Problem Statement

Determine the critical numbers for the function \( f(x)=x^{1/3}-x^{-2/3} \).

Solution

Krista King Math - 1333 video solution

video by Krista King Math

Final Answer

\(x=0\) and \(x=-2\)

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\(\displaystyle{f(x)=\frac{x^2}{x^2-9}}\)

Problem Statement

Determine the critical numbers for \(\displaystyle{f(x)=\frac{x^2}{x^2-9}}\).

Solution

PatrickJMT - 1334 video solution

video by PatrickJMT

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\(\displaystyle{g(x)=x\sqrt{16-x^2}}\)

Problem Statement

Determine the critical numbers for \(\displaystyle{g(x)=x\sqrt{16-x^2}}\).

Solution

PatrickJMT - 1335 video solution

video by PatrickJMT

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Practice Instructions

Unless otherwise instructed, determine the critical numbers of these functions.

Do NOT follow this link or you will be banned from the site!

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