Problem 13 - Precalculus: Functions and Graphs

Question

Find all values of \(x\) such that \(f(x)>0\) and all \(x\) such that \(f(x)<0,\) and sketch the graph of \(f\) $$f(x)=\frac{1}{4} x^{3}-2$$

Step-by-Step Solution

Answer

\( f(x) > 0 \) for \( x > 2 \), \( f(x) < 0 \) for \( x < 2 \).

Step 1: Understanding the Function

We're given the function \( f(x) = \frac{1}{4}x^3 - 2 \). This is a cubic function shifted downward by 2 units. We need to determine when this function is greater than zero (\( f(x) > 0 \)) and when it is less than zero (\( f(x) < 0 \)).

Step 2: Set the Function Equal to Zero

To find when \( f(x) = 0 \), we solve the equation \( \frac{1}{4}x^3 - 2 = 0 \). Adding 2 to both sides gives \( \frac{1}{4}x^3 = 2 \).

Step 3: Solve for x

Multiply both sides by 4 to get rid of the fraction: \( x^3 = 8 \). Take the cube root of both sides to solve for \( x \): \( x = \sqrt[3]{8} = 2 \). Thus, \( f(x) = 0 \) when \( x = 2 \).

Step 4: Determine when f(x) > 0

The function \( f(x) = \frac{1}{4}x^3 - 2 > 0 \) when \( \frac{1}{4}x^3 > 2 \). This simplifies to \( x^3 > 8 \), which implies \( x > 2 \).

Step 5: Determine when f(x) < 0

The function \( f(x) = \frac{1}{4}x^3 - 2 < 0 \) when \( \frac{1}{4}x^3 < 2 \). This simplifies to \( x^3 < 8 \), which implies \( x < 2 \).

Step 6: Sketch the Function

The function \( f(x) = \frac{1}{4}x^3 - 2 \) is a cubic curve shifted downwards by 2 units. It has an inflection point at the origin and crosses the x-axis at \( x = 2 \). The graph rises on the right (as \( x \rightarrow \infty \)) and falls on the left (as \( x \rightarrow -\infty \)). This indicates \( f(x) < 0 \) for \( x < 2 \) and \( f(x) > 0 \) for \( x > 2 \).

Step 7: Confirm and Conclude

From our analysis, \( f(x) > 0 \) for \( x > 2 \) and \( f(x) < 0 \) for \( x < 2 \). The sketch confirms the behavior of the function.

Key Concepts

Graphing Cubic Functions

Graphing a cubic function involves understanding the basic shape and transformations that apply to the equation. For the function given, \[ f(x) = \frac{1}{4}x^3 - 2 \], the term \( \frac{1}{4}x^3 \) represents a standard cubic curve which is typically shaped like a stretched "S".
When you

• Multiply by \( \frac{1}{4} \), it vertically compresses the function, making it less steep compared to \( x^3 \).
• Subtract 2, the entire graph shifts downward by 2 units, moving the curve below the x-axis initially.

To graph this function, plot points for different values of \( x \) and observe the smooth curve that reflects these transformations.
Notice also the inflection point at the origin. This point indicates where the curvature of the graph changes direction. The function crosses the x-axis at the root \( x = 2 \). Study the parts where the function is above or below the x-axis to understand where it’s positive or negative.

Understanding Inequalities in Cubic Functions

Inequalities help us determine the intervals where a function is positive or negative. Given the cubic function \[ f(x) = \frac{1}{4}x^3 - 2 \], you will want to find where it is greater than zero, \( f(x) > 0 \), and less than zero, \( f(x) < 0 \).
To solve this:

• Set the function equal to zero and solve \( \frac{1}{4}x^3 - 2 = 0 \) to get \( x = 2 \). This tells us where the function crosses the x-axis.
• To find \( f(x) > 0 \), examine the inequality \( x^3 > 8 \), which simplifies to \( x > 2 \). This means the function is positive for any \( x \) greater than 2.
• For \( f(x) < 0 \), look at \( x^3 < 8 \), translating to \( x < 2 \). Therefore, the function is negative for any \( x \) less than 2.

By analyzing these inequalities, you identify the qualitative behavior of cubic functions in different intervals of \( x \).

Function Analysis of Cubic Functions

Analyzing a cubic function like \[ f(x) = \frac{1}{4}x^3 - 2 \] involves several key steps to understand its overall behavior and critical points.
First, consider transformations: the term \( -2 \) causes the overall function to shift down. This means the starting point of analysis considers this vertical shift.
Then, look for roots by solving \( f(x) = 0 \). The root \( x = 2 \) is crucial as it divides the function into regions of positivity and negativity.

• Before \( x = 2 \), the function decreases and remains negative. As you move from negative infinity towards \( x = 2 \), \( f(x) \) stays below the x-axis.
• After \( x = 2 \), \( f(x) \) becomes positive, indicating an increase as \( x \) heads towards positive infinity.

Lastly, observe the behavior at extremes: As \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \), and as \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \). This indicates a typical cubic growth pattern where the ends of the graph move in opposite vertical directions.