Problem 116 - Thomas Calculus in SI Units

Question

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. How are the values at which these graphs intersect the \(x\) -axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? $$y=\frac{x^{4}}{4}-\frac{x^{3}}{3}-4 x^{2}+12 x+20$$

Step-by-Step Solution

Answer

Critical points: Solve \(x^3 - x^2 - 8x + 12 = 0\). Inflection points: Solve \(3x^2 - 2x - 8 = 0\).

Step 1: Find the First Derivative

To find the local maxima and minima, we first need the first derivative of the function. Differentiate the function: \[ y = \frac{x^4}{4} - \frac{x^3}{3} - 4x^2 + 12x + 20 \]\[ y' = x^3 - x^2 - 8x + 12 \].

Step 2: Identify Critical Points

Set the first derivative equal to zero to find critical points:\[ x^3 - x^2 - 8x + 12 = 0 \]Solve this equation for \(x\). The critical points are the roots of this polynomial.

Step 3: Use Second Derivative Test

Calculate the second derivative to determine the nature of each critical point:\[ y'' = 3x^2 - 2x - 8 \].Substitute each critical point from Step 2 into \(y''\) to determine if the point is a local maximum, local minimum, or neither.

Step 4: Find Inflection Points

Set the second derivative equal to zero and solve for \(x\) to find potential inflection points:\[ 3x^2 - 2x - 8 = 0 \].Solve this quadratic for \(x\). Check these points to confirm they are inflection points by testing the change in concavity.

Step 5: Graph Function and Derivatives

Graph the original function, its first derivative, and its second derivative. Make sure the graph is detailed enough to show all the critical points and inflection points found.

Step 6: Analyze Intersections with X-axis

Compare where the first derivative and second derivative graphs intersect the x-axis with the original function's graph. Intersections of the first derivative with the x-axis correspond to maxima or minima of the original function, while intersections of the second derivative can be potential inflection points.

Step 7: Analyze Relationship Between Graphs

Examine how the graph of the function changes: where the first derivative is zero is where the original function either stops increasing or decreasing, creating a peak (local maxima) or trough (local minima); where the second derivative changes sign indicates a change in concavity, marking an inflection point.

Key Concepts

First Derivative

The first derivative of a function gives us valuable information about its rate of change. For the function \[ y = \frac{x^4}{4} - \frac{x^3}{3} - 4x^2 + 12x + 20 \], the first derivative is \[ y' = x^3 - x^2 - 8x + 12 \]. By finding this derivative, we learn about the slope of the function at any point. This is crucial because:

• Positive derivative: The function is increasing.
• Negative derivative: The function is decreasing.

Critical points happen where the derivative is zero or undefined, indicating potential peaks or troughs.
To locate these points, set \[ y' = 0 \], solving gives the critical points.
They are our key to understanding the local behavior of the function.

Second Derivative

The second derivative tells us about the concavity of the function. For our example function, the second derivative is:\[ y'' = 3x^2 - 2x - 8 \].It helps to determine what is happening at each critical point found using the first derivative.

• If \( y'' > 0 \), the function is concave up, indicating a local minimum.
• If \( y'' < 0 \), the function is concave down, indicating a local maximum.

By setting \[ y'' = 0 \], we find the potential inflection points where the concavity changes. Inflection points are significant as they suggest a point where behavior of the graph shifts.

Local Maximum and Minimum

The local maximum and minimum of a function occur at the critical points identified using the first derivative.
After determining the critical points, we use the second derivative to ascertain whether each one is a maximum or minimum. This process is known as the second derivative test:

• If \( y'' > 0 \), the point is a local minimum, the curve bottoms out here.
• If \( y'' < 0 \), the point is a local maximum, the curve tops out here.

These local extremes give insights into the function's behavior around these points and are important for fully understanding the graph.
Local maximum and minimum values are the function's highs and lows within a certain interval, but not necessarily throughout the entire domain.

Graphing Functions

Graphing functions involves plotting the original function and its derivatives on a coordinate plane. This visual representation helps see the patterns and behaviors clearly.

• The original function's graph shows overall trends, such as increasing or decreasing behavior.
• The first derivative's graph crosses the x-axis at critical points, where the original function's slope is zero.
• The second derivative's graph helps identify potential inflection points where the graph's concavity changes.

Include all critical and inflection points in your graph. This ensures all key features of the function are visible, making it easier to understand the underlying mathematics.
Seeing these graphs together unveils how derivatives inform the shape and behavior of the original function.

Critical Points

Critical points are vital to analyzing a function's graph. They occur where the first derivative is zero, indicating potential hills or valleys in the graph.

• Find these by solving \( y' = 0 \).
• Use the second derivative test at each critical point to determine if they are maxima or minima.

Beyond just zeros of the derivative, critical points can also occur when the derivative doesn't exist, though this is less common in polynomial functions like ours.
Understanding critical points helps us delineate where the function changes its increasing/decreasing behavior, and plays a significant role in sketching accurate graphs that visually communicate the function's behavior.