Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=3 x^{4}-6 x^{2}\)

Step-by-Step Solution

Answer

The function \(f(x) = 3x^4 - 6x^2\) increases on the interval \((-∞, -1)\) and \((-1, 0)\) and \((1, ∞)\), and decreases on the interval \((0, 1)\). The behavior of the function changes at \(x = -1, 0, 1\).

Step 1: Compute the first derivative

The first derivative of the function \(f(x) = 3x^4 - 6x^2 \) can be determined using the power rule, which gives us \(f'(x) = 12x^3 - 12x = 12x(x^2 - 1)\).

Step 2: Find critical points

Critical points occur where the derivative is zero or undefined. In this case, \(f'(x) = 0\) when \(x = 0\) or \(x = 1\) or \(x = -1\). Therefore, the critical points are at \(x = 0\), \(x = 1\), and \(x = -1\).

Step 3: Determine where the function increases or decreases

We can now determine where the function increases or decreases by testing the sign of the derivative on the intervals defined by the critical points. For \(x < -1, f'(x) > 0\), thus the function is increasing. For \(-1 < x < 0, f'(x) > 0\), the function is increasing. For \(0 < x < 1, f'(x) < 0\), the function is decreasing. For \(x > 1, f'(x) > 0\), the function is increasing again.

Key Concepts

First Derivative

The first derivative of a function provides valuable information about the function's rate of change. In calculus, the first derivative refers to the slope of the tangent line to a curve at any given point. For the function \( f(x) = 3x^4 - 6x^2 \), the first derivative is obtained using differentiation rules.

To calculate the first derivative, we use the power rule, which will be explained in more detail later. Applying the power rule to each term, we find that \( f'(x) = 12x^3 - 12x \). This derivative represents how the function \( f(x) \) changes as \( x \) increases or decreases.

Finding the first derivative is a crucial step, as it lays the foundation for determining critical points and understanding the general behavior of the function.

Critical Points

Critical points of a function occur where the first derivative is either zero or undefined. These points are important because they represent places where the function's behavior could change, such as turning from increasing to decreasing or vice versa.

For the function \( f(x) = 3x^4 - 6x^2 \), the first derivative is \( f'(x) = 12x(x^2 - 1) \). By setting this derivative equal to zero, we can find the critical points. Solving \( 12x(x^2 - 1) = 0 \) gives us the roots \( x = 0 \), \( x = 1 \), and \( x = -1 \). These critical points divide the number line into intervals, each of which describes a different behavior of the function.

Critical points are essential for analyzing how functions behave over certain intervals and help identify maxima, minima, or potential inflection points.

Increasing and Decreasing Behavior

The increasing and decreasing nature of a function tells us where the function rises or falls as \( x \) moves along the x-axis. We can determine this behavior using the first derivative. If \( f'(x) > 0 \), the function is increasing. If \( f'(x) < 0 \), the function is decreasing.

For the function \( f(x) = 3x^4 - 6x^2 \), the critical points \( x = -1 \), \( x = 0 \), and \( x = 1 \) allow us to define intervals: \( x < -1 \), \( -1 < x < 0 \), \( 0 < x < 1 \), and \( x > 1 \).

• For \( x < -1 \): \( f'(x) > 0 \), hence the function is increasing.
• For \( -1 < x < 0 \): \( f'(x) > 0 \), so the function continues to increase.
• For \( 0 < x < 1 \): \( f'(x) < 0 \), indicating the function is decreasing.
• For \( x > 1 \): \( f'(x) > 0 \), meaning the function is back to increasing.

Figuring out these behaviors helps us understand the graph's shape and trajectory.

Power Rule

The power rule is a fundamental technique in calculus for finding derivatives easily. It's a formula used to differentiate expressions of the form \( x^n \). According to the power rule, if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

In the equation \( f(x) = 3x^4 - 6x^2 \), we apply the power rule to each term to find the first derivative. For the term \( 3x^4 \), applying the power rule gives us \( 12x^3 \). For \( -6x^2 \), the power rule results in \( -12x \). Thus, the first derivative is \( f'(x) = 12x^3 - 12x \).

• The power rule simplifies calculations by reducing lengthy algebraic work.
• It's specifically useful for polynomials, making it a handy tool in calculus.

Mastering the power rule is essential for performing more advanced calculus operations effectively.