Problem 243 - Calculus Volume 1

Question

Find the \(x\) -coordinates at which the tangent line to \(y=\left(x-\frac{6}{x}\right)^{8}\) is horizontal.

Step-by-Step Solution

Answer

The tangent line is horizontal at \(x = \sqrt{6}\) and \(x = -\sqrt{6}\).

Step 1: Differentiate the function

To find where the tangent line is horizontal, we need the derivative to be zero. Start by differentiating the function. Let \( y = \left( x - \frac{6}{x} \right)^8 \). Use the chain rule. \( y' = 8\left( x - \frac{6}{x} \right)^7 \cdot \left( 1 + \frac{6}{x^2} \right) \). Simplify the derivative.

Step 2: Simplify the derivative

Simplify the expression from Step 1: \[ y' = 8\left( x - \frac{6}{x} \right)^7 \left(1 + \frac{6}{x^2}\right) = 8\left( x - \frac{6}{x} \right)^7 \left(\frac{x^2 + 6}{x^2}\right) \]This is the simplified derivative expression.

Step 3: Set the derivative to zero

For horizontal tangents, set the derivative equal to zero. \[ 8\left( x - \frac{6}{x} \right)^7 \left(\frac{x^2 + 6}{x^2}\right) = 0 \]The derivative is zero when \( x - \frac{6}{x} = 0 \) or when the numerator of the second fraction \((x^2 + 6) = 0\).

Step 4: Solve for \(x\) when \( x - \frac{6}{x} = 0 \)

Solve \( x - \frac{6}{x} = 0 \) to find the \(x\)-coordinates:\[ x = \frac{6}{x} \]Multiply both sides by \(x\): \[ x^2 = 6 \]So, \( x = \sqrt{6} \) or \( x = -\sqrt{6} \).

Step 5: Check \(x^2 + 6 = 0\) for solutions

The equation \(x^2 + 6 = 0\) has no real solutions because \(x^2 = -6\) cannot happen for real numbers since square numbers are always non-negative.

Key Concepts

Derivatives

In calculus, derivatives are fundamental in understanding the rate of change of a function. For a function defined as \( y = f(x) \), the derivative, denoted as \( y' \) or \( f'(x) \), provides insight into how \( y \) changes with respect to changes in \( x \).
In practical terms, derivatives allow us to find slopes of tangent lines, determine increasing or decreasing behavior, and analyze curvature.
In our problem, we differentiate the function \( y = \left( x - \frac{6}{x} \right)^8 \), which is critical to finding where the tangent line is horizontal. Different differentiation rules, like the chain rule, can be particularly helpful when dealing with complex functions.

Tangent Lines

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point, without crossing over. The slope of this tangent line is precisely the derivative of the function at that point.
When we talk about horizontal tangent lines, we are interested in locations where the slope of the tangent line—therefore, the derivative—is exactly zero.
This means that at these points, the curve neither rises nor falls, which is like a flat line. Solving for these points enables us to understand where the function's rate of change transitions, a critical step in understanding the function's behavior.

Chain Rule

The chain rule is an essential differentiation technique for dealing with composite functions. A composite function is like \( f(g(x)) \), where the chain rule essentially states that the derivative of \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).
In our given exercise, the function \( y = \left( x - \frac{6}{x} \right)^8 \) is a composite function. Here, \( g(x) = x - \frac{6}{x} \) is nested inside \( f(x) = x^8 \).
Applying the chain rule helps us differentiate these types of functions effectively, showing us how internal and external changes combine to affect the overall rate of change of the function.

Horizontal Tangents

Horizontal tangents correspond to points on the curve where the slope of the tangent is zero. This implies that the derivative of the function at these points is zero.
To find these points, you solve the equation involving the derivative set to zero, \( y' = 0 \).
For the function \( y = \left( x - \frac{6}{x} \right)^8 \), horizontal tangent points are found by solving \( x - \frac{6}{x} = 0 \), leading us to the specific \( x \)-values where the slope vanishes, and the function momentarily "flattens out".

Solving Equations

Solving equations is a critical part of finding specific characteristics of functions, such as horizontal tangents. After setting the derivative to zero, you typically find one or more equations to solve for \( x \)-values.
In our problem, using \( x - \frac{6}{x} = 0 \), we multiply through by \( x \) to clear the fraction and then solve \( x^2 = 6 \).
This provides solutions \( x = \sqrt{6} \) and \( x = -\sqrt{6} \), which are the critical points where the tangent line to the original function is horizontal. Recognizing and applying strategies for solving equations effectively allows us to find these points.