(a) Find The Critical Numbers Of (if Any), (b) Find The Open Interval(s) On Which The Function Is Increasing
Understanding the behavior of functions is fundamental in calculus, especially when analyzing how functions change over their domains. Two essential concepts in this context are critical numbers and intervals of increase. Critical numbers help identify potential points where the function's behavior might change, such as local maxima, minima, or points of inflection. Intervals of increase, on the other hand, show where the function is rising as you move along the x-axis, providing insight into the function's growth patterns.
In this comprehensive guide, we will explore how to find the critical numbers of a function and determine the open intervals where the function is increasing. We will discuss the theoretical foundations, step-by-step procedures, common pitfalls, and practical examples to solidify your understanding.
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Understanding Critical Numbers
What Are Critical Numbers?
Critical numbers (or critical points) of a function are the values of \( x \) in the domain where the derivative of the function is either zero or undefined. These points are significant because they are potential locations of local maxima, minima, or saddle points (points of inflection).
Formally, suppose \( f \) is a function differentiable on an open interval \( I \). The critical numbers of \( f \) are all \( c \in I \) such that:
- \( f'(c) = 0 \), or
- \( f'(c) \) does not exist.
Critical numbers serve as the starting point for analyzing the function's increasing/decreasing behavior and for locating extrema.
Why Are Critical Numbers Important?
- They help identify potential local extrema.
- They are crucial in applying the First and Second Derivative Tests.
- They assist in sketching the graph of a function by highlighting key points where the function's slope changes.
How to Find Critical Numbers
The process involves the following steps:
- Find the derivative \( f'(x) \) of the function.
- Solve the equation \( f'(x) = 0 \) to find where the slope is zero.
- Find where \( f'(x) \) is undefined, provided these points are in the domain.
- Collect all such \( x \)-values as critical numbers.
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Step-by-Step Procedure for Finding Critical Numbers
Step 1: Differentiate the Function
Begin by computing the derivative \( f'(x) \). Depending on the function's complexity, this may involve applying rules such as the power rule, product rule, quotient rule, or chain rule.
Example: For \( f(x) = x^3 - 3x^2 + 2 \), the derivative is \( f'(x) = 3x^2 - 6x \).
Step 2: Solve \( f'(x) = 0 \)
Set the derivative equal to zero and solve for \( x \):
\[ 3x^2 - 6x = 0 \]
\[ 3x(x - 2) = 0 \]
\[ x = 0 \quad \text{or} \quad x = 2 \]
These are potential critical numbers.
Step 3: Find Points Where \( f'(x) \) Is Undefined
Identify points where the derivative does not exist, such as points involving division by zero or discontinuities.
Example: For \( f(x) = \frac{1}{x - 1} \), the derivative exists everywhere except at \( x = 1 \), where the function is undefined. So, \( x=1 \) is a critical number.
Step 4: Verify Critical Numbers Are in the Domain
Ensure that the critical numbers found are within the domain of the original function. If a critical number falls outside the domain, it is not valid for analysis.
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Finding Intervals Where the Function Is Increasing
Understanding Intervals of Increase
A function \( f \) is said to be increasing on an interval \( (a, b) \) if for any two points \( x1, x2 \in (a, b) \) with \( x1 < x2 \), we have:
\[ f(x1) < f(x2) \]
Graphically, the function rises as we move from left to right within this interval.
Role of the Derivative in Determining Intervals of Increase
The First Derivative Test states:
- If \( f'(x) > 0 \) for every \( x \) in an interval, then \( f \) is increasing on that interval.
- If \( f'(x) < 0 \) for every \( x \) in an interval, then \( f \) is decreasing there.
Thus, analyzing the sign of the derivative helps identify where the function increases or decreases.
Procedure to Find Intervals of Increase
- Identify Critical Numbers: As above, find all \( c \) where \( f'(c) = 0 \) or \( f'(c) \) is undefined.
- Partition the Domain: Using the critical numbers as partition points, divide the domain into intervals.
- Test the Sign of \( f'(x) \): Pick test points within each interval and evaluate \( f'(x) \).
- Determine the Behavior:
- If \( f'(x) > 0 \) on an interval, \( f \) is increasing there.
- If \( f'(x) < 0 \), \( f \) is decreasing.
Applying the Concepts Through Examples
Example 1: Finding Critical Numbers and Intervals of Increase for a Polynomial Function
Given: \( f(x) = x^3 - 6x^2 + 9x + 2 \)
Step 1: Derive \( f'(x) \):
\[ f'(x) = 3x^2 - 12x + 9 \]
Step 2: Set \( f'(x) = 0 \):
\[ 3x^2 - 12x + 9 = 0 \]
\[ x^2 - 4x + 3 = 0 \]
\[ (x - 1)(x - 3) = 0 \]
\[ x = 1 \quad \text{or} \quad x = 3 \]
Step 3: Critical numbers are \( x=1 \) and \( x=3 \).
Step 4: Partition the domain into intervals:
- \( (-\infty, 1) \)
- \( (1, 3) \)
- \( (3, \infty) \)
Step 5: Test \( f'(x) \) in each interval:
- For \( x=0 \) in \( (-\infty, 1) \):
\[ f'(0) = 3(0)^2 - 12(0) + 9 = 9 > 0 \]
- For \( x=2 \) in \( (1, 3) \):
\[ f'(2) = 3(4) - 12(2) + 9 = 12 - 24 + 9 = -3 < 0 \]
- For \( x=4 \) in \( (3, \infty) \):
\[ f'(4) = 3(16) - 12(4) + 9 = 48 - 48 + 9 = 9 > 0 \]
Conclusion:
- \( f \) is increasing on \( (-\infty, 1) \) and \( (3, \infty) \).
- \( f \) is decreasing on \( (1, 3) \).
Example 2: Analyzing a Rational Function
Given: \( g(x) = \frac{x^2 - 4}{x - 1} \)
Step 1: Find derivative \( g'(x) \):
Using the quotient rule:
\[ g'(x) = \frac{(2x)(x-1) - (x^2 - 4)(1)}{(x-1)^2} \]
Simplify numerator:
\[ 2x(x-1) - (x^2 - 4) = 2x^2 - 2x - x^2 + 4 = x^2 - 2x + 4 \]
Thus,
\[ g'(x) = \frac{x^2 - 2x + 4}{(x - 1)^2} \]
Step 2: Find critical points:
- Numerator \( x^2 - 2x + 4 \):
Discriminant: \( (-2)^2 - 4 \times 1 \times 4 = 4 - 16 = -12 < 0 \)
Since the numerator is always positive (quadratic with positive leading coefficient and negative discriminant), \( g'(x) \) is never zero.
- The derivative is undefined at \( x=1 \). Since \( x=1 \) is not in the domain (denominator zero), it is a vertical asym
