Problem 14 - Calculus

Question

The slope of \(x+(1 / x)\) is zero when \(x=\) _______ .What does the graph do at that point?

Step-by-Step Solution

Answer

The slope is zero at \( x = 1 \) and \( x = -1 \). The graph has horizontal tangents at these points.

Step 1: Define the function

The function given is \( f(x) = x + \frac{1}{x} \). Our goal is to find where the slope of this function is zero.

Step 2: Find the derivative

The derivative of the function \( f(x) = x + \frac{1}{x} \) is \( f'(x) = 1 - \frac{1}{x^2} \). This derivative represents the slope of the function at any point \( x \).

Step 3: Set the derivative equal to zero

To find where the slope is zero, set the derivative equal to zero: \( 1 - \frac{1}{x^2} = 0 \).

Step 4: Solve for \( x \)

Rearrange the equation from Step 3 to solve for \( x \): \( 1 = \frac{1}{x^2} \). Thus, \( x^2 = 1 \), which gives us \( x = 1 \) or \( x = -1 \).

Step 5: Analyze the critical points

At \( x = 1 \) and \( x = -1 \), the slope of the function is zero, indicating that these are critical points where the graph has horizontal tangents, typically a minimum or a maximum.

Key Concepts

Critical Points

Critical points in calculus are places on the graph of a function where certain interesting behaviors occur. These are typically points where the derivative of the function equals zero or where the derivative does not exist. For the function in question,

• The derivative was found to be zero at specific values of \( x \), namely \( x=1 \) and \( x=-1 \).
• These values are what we call critical points.

At these points, the function’s slope becomes zero, which often means the graph has a horizontal tangent. This can indicate a potential minimum or maximum on the graph. Identifying critical points helps in analyzing the overall behavior and structure of a function.

Derivative of a Function

The derivative of a function provides insight into the function's rate of change. It's essentially like taking a peek into where the function is heading—increasing or decreasing.
For any function \( f(x) \), the derivative \( f'(x) \) tells us:

• Where the function is rising or falling, and how steep that slope is at any given point.
• The value of the derivative at a certain point is an immediate measure of the slope of the tangent to the curve at that point.

In our example with \( f(x) = x + \frac{1}{x} \), differentiating leads us to \( f'(x) = 1 - \frac{1}{x^2} \). This function describes the slope of \( f(x) \) at every point \( x \). Derivatives are key in understanding the dynamics of a function’s graph.

Zero Slope

A zero slope on the graph of a function indicates a flat or horizontal tangent at that point. This occurs because the derivative, which determines the slope, equals zero. In practical terms:

• Having a zero slope means the function is neither increasing nor decreasing at that particular point.
• This situation is crucial in determining the critical points, as seen when \( f'(x) = 1 - \frac{1}{x^2} = 0 \), leading to solutions \( x=1 \) and \( x=-1 \).

Zero slope points are often investigated further to see if they correspond to local maxima, minima, or saddle points on the function's graph.

Function Analysis

Function analysis is about understanding the overall behavior and characteristics of a function. This includes exploring its increases, decreases, and the potential locations where it hits its highest or lowest points.
For the function \( f(x) = x + \frac{1}{x} \):

• We found critical points at \( x=1 \) and \( x=-1 \) where the slope is zero using its derivative \( f'(x) = 1 - \frac{1}{x^2} \).
• This information is then used to determine the nature of these points—are they local maxima, minima, or something else?
• Analyzing the graph of the function further can help clarify its behavior between these critical points, giving a full view of the function’s tendencies.

Engaging in function analysis uses calculus tools, like derivatives, to produce a map of the function's highs and lows across its domain.