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Chapter 3: Problem 15

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

Short Answer

Expert verified
For \(x > 2\), \(f(x) > 0\). For \(x < 2\), \(f(x) < 0\). The graph is a cubic curve crossing the x-axis at \(x=2\).

Step by step solution

01

Understand the Function

The function given is \( f(x) = \frac{1}{4}x^3 - 2 \). It is a cubic function which typically has an S-shape. We need to find where this function is positive (\( f(x) > 0 \)) and where it is negative (\( f(x) < 0 \)).
02

Solve the Inequality \(f(x) > 0\)

Set the inequality \( \frac{1}{4}x^3 - 2 > 0 \). Add 2 to both sides: \( \frac{1}{4}x^3 > 2 \). Multiply both sides by 4 to clear the fraction: \( x^3 > 8 \). Take the cube root: \( x > 2 \). Thus, \( f(x) \) is positive for \( x > 2 \).
03

Solve the Inequality \(f(x) < 0\)

Set the inequality \( \frac{1}{4}x^3 - 2 < 0 \). Add 2 to both sides: \( \frac{1}{4}x^3 < 2 \). Multiply both sides by 4: \( x^3 < 8 \). Take the cube root: \( x < 2 \). Thus, \( f(x) \) is negative for \( x < 2 \).
04

Combine the Solutions

From the previous steps, we found that \( f(x) > 0 \) when \( x > 2 \) and \( f(x) < 0 \) when \( x < 2 \). At \( x = 2 \), \( f(x) = 0 \).
05

Sketch the Graph of \(f(x)\)

Plot the critical point where \( x = 2 \), \( f(x) = 0 \). The graph of \( f(x) \) is a cubic graph and passes through \( y = -2 \) when \( x = 0 \). It crosses the x-axis at \( x = 2 \) because this is where \( f(x) = 0 \). For \( x < 2 \), the graph is below the x-axis (negative), and for \( x > 2 \), it is above the x-axis (positive). Draw a smooth S-shaped curve reflecting these behaviors.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequalities
When dealing with inequalities involving a cubic function like \(f(x) = \frac{1}{4}x^3 - 2\), we seek to determine the intervals on which the function is positive or negative. Specifically:
  • \(f(x) > 0\): Here, we want to find all \(x\) such that the function yields a positive result. This inequality signifies that the cubic graph lies above the x-axis.
  • \(f(x) < 0\): In this case, \(x\) values should produce a negative \(f(x)\), meaning the graph is below the x-axis.
Remember, solving these inequalities often involves isolating the variable in question. For the function given, this required manipulating the inequality to \(x^3 > 8\) and \(x^3 < 8\). Each inequality describes a range of values that \(x\) can take to ensure \(f(x)\) meets the condition of being greater or less than zero.
By taking cube roots, we find the critical points at \(x = 2\), allowing us to conclude that:
  • \(f(x) > 0\) for \(x > 2\)
  • \(f(x) < 0\) for \(x < 2\)
Graph Sketching
Graph sketching is a helpful tool to visually understand the behavior of functions across their domains. For a cubic function, we expect the graph to have an "S" shape due to its third-degree polynomial nature.When sketching the graph of \(f(x) = \frac{1}{4}x^3 - 2\), note the following:
  • The function crosses the x-axis at \(x = 2\), where \(f(x) = 0\). This point is crucial as it divides the graph into the regions where the function is positive and negative.
  • At \(x = 0\), the function value is \(f(x) = -2\), providing another point to help anchor the graph below the x-axis initially.
  • The graph transitions from negative to positive as it moves from left to right across \(x = 2\).
By marking these critical points and understanding their roles, you can sketch a smooth curve that dips below the x-axis for \(x < 2\) and rises above it for \(x > 2\). The graph represents the natural curve of the cubic function as it approaches, intersects, and moves past the x-axis.
Polynomial Functions
Polynomial functions are amongst the fundamental building blocks in algebra, characterized by terms that involve powers of \(x\), with coefficients. The function \(f(x) = \frac{1}{4}x^3 - 2\) is a specific type of polynomial, known as a cubic polynomial, due to its highest power being three.Key points to note:
  • The leading term, \(\frac{1}{4}x^3\), dictates the function's end behavior. For large positive or negative \(x\), the function will increase or decrease without bound due to this term.
  • The constant term, \(-2\), shifts the entire graph up or down on the y-axis.
  • Cubic polynomials can have up to three x-intercepts, but in this case, we have one real root at \(x = 2\).
Polynomial functions like this are smooth and continuous, meaning there are no breaks, holes, or jumps in their graphs. Understanding the impact of each term helps in predicting and sketching the overall shape of the function. This insight into polynomial behavior is fundamental to advancing in algebra and calculus.

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