Problem 98 - Calculus and its applications

Question

Find the interval(s) for which \(f^{\prime}(x)\) is positive. Find the points on the graph of $$ y=2 x^{6}-x^{4}-2 $$ at which the tangent line is horizontal.

Step-by-Step Solution

Answer

The points are \((0, -2)\), \((\sqrt{\frac{1}{3}}, -\frac{10}{9})\), and \((-\sqrt{\frac{1}{3}}, -\frac{10}{9})\)."

Step 1: Differentiate the Function

First, we need to find the first derivative of the function to determine where its slopes (tangent lines) are horizontal. The function is given by \( y = 2x^6 - x^4 - 2 \). Using the power rule, the first derivative is:\( f^{\prime}(x) = \frac{d}{dx}(2x^6) - \frac{d}{dx}(x^4) - \frac{d}{dx}(2). \)Calculating each term separately:1. \( \frac{d}{dx}(2x^6) = 12x^5 \),2. \( \frac{d}{dx}(x^4) = 4x^3 \),3. \( \frac{d}{dx}(2) = 0. \)Thus, the first derivative of the function is\( f^{\prime}(x) = 12x^5 - 4x^3. \)

Step 2: Set First Derivative to Zero

To find where the tangent line is horizontal, we set the first derivative \( f^{\prime}(x) \) equal to zero and solve for \( x \):\( 12x^5 - 4x^3 = 0. \) Factor out the common term \( 4x^3 \):\( 4x^3(3x^2 - 1) = 0. \) This gives us two equations to solve: \( 4x^3 = 0 \) and \( 3x^2 - 1 = 0. \)\Solving \( 4x^3 = 0 \) gives \( x = 0. \)Solving \( 3x^2 - 1 = 0 \) gives \( x^2 = \frac{1}{3} \), so \( x = \pm\sqrt{\frac{1}{3}}. \)

Step 3: Find Tangent Line Points

Now, we need to find the corresponding \( y \)-values for each solution we found in the previous step. These points will have horizontal tangent lines:1. For \( x = 0 \): \( y = 2(0)^6 - (0)^4 - 2 = -2. \) Thus, the point is \((0, -2)\).2. For \( x = \sqrt{\frac{1}{3}} \) and \( x = -\sqrt{\frac{1}{3}} \): - Substitute into \( y = 2x^6 - x^4 - 2 \) to find \( y \). - These points are \((\sqrt{\frac{1}{3}}, -\frac{10}{9})\) and \((-\sqrt{\frac{1}{3}}, -\frac{10}{9})\).

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus, representing how a function changes as its input changes. You can think of a derivative as the "rate of change" or "slope" at any given point on a function. For example, if you have a function describing the height of a ball being thrown into the air, the derivative would tell you how fast the height changes as time goes on.

Finding a derivative involves applying certain rules to a function. The end goal is to determine the function's slope at any given point. In other terms:

• A derivative indicates where a function is increasing or decreasing.
• If the derivative is positive, the function is increasing.
• If the derivative is negative, the function is decreasing.

This information can be used to identify intervals where the function behaves in a certain way. Derivatives play a crucial role in many applications, from physics to economics, by providing insights into how various conditions affect dynamic systems.

Horizontal Tangent Lines

A tangent line is a straight line that just "touches" a curve at a specific point. When a tangent line is horizontal, it means that the slope of the function at that point is zero. In simpler terms, the function isn't going up or down at that exact spot.

• The slope of the tangent line (and thus the first derivative) is zero.
• These points are often critical points where the function might change from increasing to decreasing, or vice versa.

To find where the tangent line to a graph is horizontal, you set the first derivative of the function equal to zero and solve for the input values. These input values are the "x" coordinates where the tangency occurs, providing valuable insights into the behavior of the function.

Once you have these "x" values, you can calculate the corresponding "y" values by substituting them back into the original function. This gives you the full coordinates of points with horizontal tangent lines. These points are essential for sketching graphs and understanding function behaviors at various intervals.

The Power Rule for Derivatives

The power rule is one of the most straightforward and widely used rules in calculus for finding derivatives. It applies when you want to differentiate a term in the form of \(x^n\), where \(n\) is a power.

According to the power rule:

• The derivative of \(x^n\) is \(nx^{n-1}\).
• In other words, multiply the entire expression by the exponent, then subtract one from the exponent to get the new power.

Applying the power rule makes it easy to differentiate polynomials. For example, when deriving terms like \(2x^6\) or \(x^4\), you simply multiply by the exponent, then decrease the exponent by one:

• For \(2x^6\), the derivative is \(12x^5\).
• For \(x^4\), the derivative is \(4x^3\).

This simple rule helps break down complex expressions into manageable parts, enabling you to calculate derivatives efficiently. Mastering the power rule is crucial for handling more advanced problems in calculus that involve finding higher-order derivatives and solving real-world application problems.