Problem 12 - Applied Calculus

Question

Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$ f(x)=3 x^{5}-5 x^{3} $$

Step-by-Step Solution

Answer

The graph of \( f(x) \) has critical points at \( x = -1, 0, \) and \( x = 1 \); it decreases on \((-1, 0)\) and \((0, 1)\), and increases on \((- fty, -1)\) and \((1, fty)\).

Step 1: Find the Derivative

First, let's find the derivative of the function \( f(x) = 3x^5 - 5x^3 \). Using the power rule, the derivative is \( f'(x) = 15x^4 - 15x^2 \).

Step 2: Factor the Derivative

Next, factor the derivative to find the critical points. The derivative \( f'(x) = 15x^4 - 15x^2 \) can be factored as \( 15x^2(x^2 - 1) = 15x^2(x - 1)(x + 1) \).

Step 3: Find the Critical Points

Set the factored derivative equal to zero to find critical points: \( 15x^2(x - 1)(x + 1) = 0 \). This gives the critical points at \( x = 0, x = 1, \) and \( x = -1 \).

Step 4: Determine Monotonic Intervals

Test intervals between the critical points to determine whether \( f(x) \) is increasing or decreasing. Test values in the intervals \((-fty, -1), (-1, 0), (0, 1), (1, fty)\) in \( f'(x) \) to find the sign of the derivative. This indicates where the function is increasing or decreasing.

Step 5: Test Values in Intervals

- For \( x = -2 \) (in \( (-fty, -1) \)), \( f'(-2) = 15 \times 4 \times (-3) > 0 \), so \( f(x) \) is increasing.- For \( x = -0.5 \) (in \( (-1, 0) \)), \( f'(-0.5) = 15 \times 0.25 \times (-0.75) < 0 \), so \( f(x) \) is decreasing.- For \( x = 0.5 \) (in \( (0, 1) \)), \( f'(0.5) = 15 \times 0.25 \times (-0.75) < 0 \), so \( f(x) \) is decreasing.- For \( x = 2 \) (in \( (1, fty) \)), \( f'(2) = 15 \times 4 \times 3 > 0 \), so \( f(x) \) is increasing.

Step 6: Analyze the Shape of the Graph

From the derivative tests, \( f(x) \) decreases on \( (-1, 0) \) and \( (0, 1) \), increases on \( (-fty, -1) \) and \( (1, fty) \), with critical points at \( x = -1, 0, 1 \). The function changes direction at these points, giving the graph a wave-like shape with local extrema.

Key Concepts

Derivative

To understand how a function behaves, we often look at its derivative. The derivative of a function gives us valuable information about the function's rate of change. Mathematically, a derivative tells us the slope of the tangent line to the graph at any given point. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing. This concept helps us analyze the behavior of functions.
In our example, the function is given by \( f(x) = 3x^5 - 5x^3 \). Using differentiation rules, specifically the power rule, we find its derivative: \( f'(x) = 15x^4 - 15x^2 \). This derivative allows us to find critical points and understand where the function increases or decreases.

Monotonic Function

A monotonic function is one that is entirely non-increasing or non-decreasing throughout its domain. This characteristic makes analyzing the function simpler, as it doesn't change direction. However, many functions, like our example \( f(x) = 3x^5 - 5x^3 \), are not monotonic over their entire domains but have segments which are monotonic.
When evaluating where a function is monotonic, we look at intervals where the derivative maintains a consistent sign. For our function, \( f(x) \) is increasing wherever \( f'(x) > 0 \) and decreasing wherever \( f'(x) < 0 \). Determining these intervals is key to understanding the overall shape of \( f(x) \).

Increasing and Decreasing Intervals

To identify where the function is increasing or decreasing, we analyze the derivative sign over different intervals. For our function \( f(x) = 3x^5 - 5x^3 \), the critical points where the derivative equals zero (\( f'(x) = 0 \)) were found at \( x = -1, 0, \) and \( 1 \). These critical points are boundaries that divide the real number line into intervals.
By testing values within these intervals in the derivative \( f'(x) = 15x^4 - 15x^2 \), we determine the behavior:

• On \( (-\infty, -1) \), \( f(x) \) is increasing.
• On \( (-1, 0) \), \( f(x) \) is decreasing.
• On \( (0, 1) \), \( f(x) \) is decreasing.
• On \( (1, \infty) \), \( f(x) \) is increasing.

This analysis provides a detailed map of the function's behavior over its domain.

Power Rule

The power rule is an essential technique in calculus for differentiating functions of the form \( x^n \). According to this rule, the derivative of \( x^n \) is given by \( nx^{n-1} \).
Using this rule significantly simplifies the process of finding derivatives for polynomial functions. In our function \( f(x) = 3x^5 - 5x^3 \), applying the power rule gives us \( f'(x) = 15x^4 - 15x^2 \). Here, each term is treated independently, showcasing the straightforward nature of this rule.
This simplicity makes the power rule a favorite among students and a powerful tool for tackling calculus problems effortlessly.