Problem 12 - Algebra and Trigonometry

Question

\(5-12\) . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f\) . (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{4}-4 x^{3}+2 x^{2}+4 x-3 $$

Step-by-Step Solution

Answer

The function increases on \((-\infty, 0.467) \cup (2.533, \infty)\) and decreases on \((0.467, 2.533)\).

Step 1: Graph the Function

Use a graphing calculator or software to plot the function \(f(x) = x^4 - 4x^3 + 2x^2 + 4x - 3\). Analyze the shape and direction of the curve within a suitable range, such as \(-5 \leq x \leq 5\). This will help in identifying the intervals where the function increases or decreases.

Step 2: Identify Critical Points

Calculate the derivative of \(f(x)\), \(f'(x) = 4x^3 - 12x^2 + 4x + 4\). Set \(f'(x) = 0\) to find the critical points, solve \(4x^3 - 12x^2 + 4x + 4 = 0\) using algebraic methods or software.

Step 3: Determine Intervals of Increase and Decrease

Use the critical points found in Step 2 to divide the x-axis into intervals. Test each interval by selecting a test point and substituting it into \(f'(x)\) to determine if the function is increasing or decreasing in that interval.

Step 4: State Intervals of Increase and Decrease

Analyze the sign of \(f'(x)\) in each interval from Step 3. If \(f'(x) > 0\), the function is increasing in that interval. If \(f'(x) < 0\), the function is decreasing. Record these intervals as your final answer.

Key Concepts

Graphing Functions

Understanding how to graph functions is a fundamental skill in calculus, providing visual insight into the behavior of functions. Functions are mathematical expressions that relate inputs to outputs, creating a graph by plotting these relationships on a coordinate plane.
When graphing the function, for instance, \(f(x) = x^4 - 4x^3 + 2x^2 + 4x - 3\), you should consider the shape and characteristics at various values of \(x\). Here’s a short guide to help you through this process:

• Choose a Range: Decide the range of \(x\)-values you will explore. In our example, the range from \(-5 \leq x \leq 5\) is a good start.
• Plot Points: Use particular \(x\)-values within your range to compute \(f(x)\) and plot these points on the graph.
• Use a Graphing Tool: While manual plotting is effective for learning, graphing software simplifies the process and provides precise graphs.
• Analyze the Curve: Assess the curve's direction and any interesting features such as peaks, valleys, or symmetry.

Graphing functions is not just about drawing; it’s about interpreting this visual data to understand where and how the function behaves.

Critical Points

Critical points in calculus are valuable because they provide information about where a function's graph might change direction. To find these points, first calculate the derivative of the function, which represents the slope of the tangent line at any point on the graph.
For \(f(x) = x^4 - 4x^3 + 2x^2 + 4x - 3\), the derivative, \(f'(x) = 4x^3 - 12x^2 + 4x + 4\), is a polynomial whose roots indicate potential critical points.
The steps for identifying critical points are:

• Find the Derivative: Use calculus techniques to compute the first derivative of the function.
• Solve for Zero: Set the derivative equal to zero, \(f'(x) = 0\), and solve for \(x\) to find critical points.
• Verify Critical Points: These points occur where the derivative equals zero or is undefined; ask if they truly indicate local maxima, minima, or inflection points by using secondary tests if needed.

Successfully identifying critical points helps in the next steps of determining where the function is increasing or decreasing.

Intervals of Increase and Decrease

Once you have identified the critical points, you can determine the intervals where the function is increasing or decreasing. This involves evaluating the sign of the derivative across these intervals.
Here's how you can do this:

• Divide the Axis: Use the critical points to section the \(x\)-axis into intervals.
• Choose Test Points: Pick a test point from each interval to substitute back into the derivative, \(f'(x)\).
• Determine the Sign: Analyze whether \(f'(x)\) is positive or negative at each test point:
• If \(f'(x) > 0\), the function is increasing on that interval.
• If \(f'(x) < 0\), the function is decreasing on that interval.
• Summarize: State the intervals, so you clearly understand the regions of increase and decrease.

By knowing these intervals, you gain a more detailed understanding of the function's behavior over its domain, essential for precise graph analysis and problem-solving in calculus.