Problem 33 - Essential Calculus

Question

Find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=\frac{e^{x}+e^{-x}}{2}\)

Step-by-Step Solution

Answer

The function \( f(x)=\frac{e^{x}+e^{-x}}{2} \) has no relative extrema as it does not have any points where the first derivative is zero.

Step 1: Calculating the first derivative

The derivative of \( f(x) \) gives us: \( f'(x)= \frac{e^{x}-e^{-x}}{2} \)

Step 2: Finding critical points

Set the derivative equal to zero to find critical values of \( x \), thus: \( 0= \frac{e^{x}-e^{-x}}{2} \). Multiplying both sides by 2 and adding \( e^{-x} \) we get \( e^{-x}=e^{x} \) which should then be solved for \( x \).

Step 3: Calculating the second derivative

Find the second derivative of \( f(x) \). The second derivative, \( f''(x) \), is \( \frac{e^{x}+e^{-x}}{2} \).

Step 4: Using the Second Derivative Test

Substitute your critical numbers from step 2 into your second derivative and check whether the results are positive, negative, or zero. This will tell you if the original function has a minimum, maximum, or an inflection point at these values of \( x \).

Key Concepts

Relative Extrema

Finding relative extrema of a function is a crucial step in understanding its overall behavior. These extrema are the highest or lowest points relative to nearby values, thus locally maximizing or minimizing the function in its immediate range.
Relative extrema help answer questions like "Where does a function peak?" or "Where is the function lowest?"
To determine these, you'll often use the second derivative test, which allows you to efficiently identify these points after finding critical points.

• Use the first derivative to find critical points, where the function's slope equals zero.
• Apply the second derivative test to determine the nature of each critical point.

Whether a relative minimum or maximum, these points provide deeper insight into the function's graph and its potential applications.

Derivatives

Derivatives play a fundamental role in calculus. They measure how a function changes at any given point, essentially acting as the "rate of change" or the "slope" of the function at a specific point.
Derivatives are critical in various fields including physics, economics, and engineering as they help predict how systems behave.
To find derivatives, we apply rules such as the power rule, product rule, and chain rule, among others, depending on the complexity of the function.

• They help us determine the slope at a point on a graph, telling us whether the function is increasing or decreasing at that point.
• First derivatives can also help identify critical points, which are candidates for relative extrema.

Through derivatives, we gain the power to analyze and predict changes, providing invaluable insights across different applications.

Critical Points

Critical points are where a function's first derivative is zero or undefined, marking potential places where the function changes direction. These points are significant as they indicate where relative extrema might occur.
Determining critical points is a two-step process:
1. Find the first derivative of the function.
2. Set the first derivative equal to zero and solve for the variable (e.g., x).
This tells us where the function's slope becomes zero—where it "flattens out." These are often locations of peaks or troughs on the function's graph.
After identifying critical points, utilize the second derivative test to ascertain if these are relative maxima or minima.

Calculus

Calculus is the mathematical study of continuous change, covering subjects such as limits, differentiation, and integration. It provides the framework for modeling dynamic systems and understanding patterns of change in the natural world, technology, and beyond.
In the context of finding relative extrema via the second derivative test, calculus helps break down the motions of functions into comprehensible units.

• It allows us to describe how quantities change together in systems.
• It provides techniques, such as the second derivative test, to further analyze and optimize functions.

From understanding the flight path of a ball to optimizing a business strategy, calculus offers the tools necessary to tackle complex questions involving change.