Problem 3 - Calculus Single Variable

Question

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=1 /\left(x^{2}+1\right) $$

Step-by-Step Solution

Answer

The function has a local maximum at x = 0.

Step 1: Understand the Function

The given function is \( f(x) = \frac{1}{x^2 + 1} \). This is a rational function, and the denominator \( x^2 + 1 \) ensures that the function is defined for all real numbers since \( x^2 + 1 > 0 \) for all \( x \).

Step 2: Consider the Nature of Rational Functions

Since \( x^2 + 1 \) is always positive, \( f(x) \) will always be positive as well. This implies that there are no zeros or undefined points for the function to analyze for critical points.

Step 3: Find Potential Extremum by Inspecting the Denominator

The expression \( x^2 + 1 \) reaches its minimum value when \( x^2 \) is minimized. Since \( x^2 \) is zero at \( x = 0 \), this is where \( f(x) \) could potentially have an extremum. Evaluate \( f(0) = \frac{1}{0^2 + 1} = 1 \).

Step 4: Analyze Function Behavior Around c=0

For any \( x eq 0 \), \( x^2 + 1 > 1 \), thus \( f(x) = \frac{1}{x^2 + 1} < 1 \). This indicates that \( x = 0 \) gives the maximum value of the function which is 1.

Step 5: Conclusion on Local Extrema

Since \( f(x) \) reaches the highest value at \( x = 0 \) and decreases as \( x \) moves away from 0, \( x = 0 \) is a local maximum for this function.

Key Concepts

Rational Functions

Rational functions are a fascinating area of study in mathematics. A rational function is defined as a function that is the ratio of two polynomials. In simpler terms,

• It can be expressed in the form \( \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials.
• In the function provided, \( f(x) = \frac{1}{x^2 + 1} \), the numerator \( 1 \) is a constant polynomial, and the denominator \( x^2 + 1 \) is a quadratic polynomial.
• Rational functions are only undefined where the denominator is zero. However, for this function, \( x^2 + 1 \) ensures the function is defined for all real numbers since it can't be zero.

Understanding rational functions allows us to graph them and analyze their behavior, especially near undefined points if they exist. In this case, the rational function is smooth and uninterrupted across its domain.

Function Behavior

The behavior of rational functions is crucial to analyze how they change over different intervals. For the function \( f(x) = \frac{1}{x^2 + 1} \), let's explore what governs its behavior.

• The denominator \( x^2 + 1 \) is always positive, meaning \( f(x) \) is always positive as well, guaranteeing no zeros (intercepts on the x-axis).
• Since there are no undefined values, \( f(x) \) is continuous for all \( x \).
• As \( x^2 \) becomes very large, \( f(x) \) approaches zero. This gives us insight into its asymptotic behavior: as \( x \) moves towards \( \pm \infty \), the function value shrinks towards zero.

Identifying these characteristics helps us understand the bigger picture, offering a snapshot of what the function looks like on a graph and predicting its changes across the domain.

Critical Points

Critical points of a function are values where the derivative is zero or undefined, potentially indicating a local maximum, minimum, or saddle point. In our context, even though calculus isn't required, understanding critical points is pivotal.

• For \( f(x) = \frac{1}{x^2 + 1} \), we identify critical points by observing where the function changes its direction of growth.
• Given that \( x = 0 \) minimizes \( x^2 \), it maximizes \( f(x) \). At \( x = 0 \), \( f(x) \) reaches its max value of 1 and can't be higher elsewhere.
• No other critical points exist because the denominator doesn't zero out and the function is smooth throughout.

Critical points like \( x = 0 \) guide us to where the function achieves significant values such as peaks, which are crucial in extremum analysis.

Extremum Analysis

Extremum refers to the maximum or minimum value a function takes on a certain interval. Understanding extremum is beneficial for identifying local high or low points within a function.

• In our function \( f(x) = \frac{1}{x^2 + 1} \), extremum analysis spotlights \( x = 0 \) as the point of highest value.
• Further analysis reveals that around \( x = 0 \), the function decreases as \( |x| \) increases, making \( x = 0 \) the local maximum.
• Beyond calculus, extremum highlights intuitive maxima or minima just by inspecting the form and behavior of the function, such as analyzing the denominator's minimum or maximum.

Understanding extremum helps in pinpointing optimal points in functions, crucial for optimization problems or even algorithm efficiency.