Problem 91 - Thomas Calculus

Question

a. On a common screen, graph \(f(x)=x^{3}+k x\) for \(k=0\) and nearby positive and negative values of \(k\) . How does the value of \(k\) seem to affect the shape of the graph? b. Find \(f^{\prime}(x)\) . As you will see, \(f^{\prime}(x)\) is a quadratic function of \(x .\) Find the discriminant of the quadratic (the discriminant of \(a x^{2}+b x+c\) is \(b^{2}-4 a c )\) . For what values of \(k\) is the discriminant positive? Zero? Negative? For what values of \(k\) does \(f^{\prime}\) have two zeros? One or no zeros? Now explain what the value of \(k\) has to do with the shape of the graph of \(f .\) c. Experiment with other values of \(k .\) What appears to happen as \(k \rightarrow-\infty ?\) as \(k \rightarrow \infty ?\)

Step-by-Step Solution

Answer

The value of \( k \) affects the graph's direction and steepness; negative \( k \) leads to more turning points, while positive \( k \) smoothens the curve.

Step 1: Understand the Graph

First, we will look at how the graph of the function changes with different values of \(k\) by graphing \( f(x) = x^3 + kx \) for \(k = 0, 1, -1, 2, -2,\) etc. For \( k = 0 \), the graph is a standard cubic curve. As \( k \) increases, the graph appears to tilt and shift, indicating steeper slopes. For negative \( k \), the curves tilts in the opposite direction.

Step 2: Find the Derivative

We find the derivative of the function \( f(x) = x^3 + kx \). Using differentiation rules, we get \( f'(x) = 3x^2 + k \). This derivative is a quadratic function of \(x\).

Step 3: Calculate the Discriminant

The discriminant of a quadratic \( ax^2 + bx + c \) is given by the formula \( b^2 - 4ac \). For our function, \( a = 3 \), \( b = 0 \), and \( c = k \), so the discriminant is \( 0^2 - 4 \cdot 3 \cdot k = -12k \).

Step 4: Analyze the Discriminant

To determine how the discriminant affects the graph, we analyze the conditions:- Positive Discriminant: \( -12k > 0 \Rightarrow k < 0 \). So, when \( k < 0 \), \( f'(x) \) has two real zeros, indicating local minima and maxima in \( f(x) \).- Zero Discriminant: \( -12k = 0 \Rightarrow k = 0 \), giving one real zero, indicating a point of inflection for \( f(x) \).- Negative Discriminant: \( -12k < 0 \Rightarrow k > 0 \), where \( f'(x) \) has no real zeros, likely having no turning points, and \( f(x) \) is monotonic.

Step 5: Explore Limits as k Changes

Experiment with extreme values:- As \( k \rightarrow -\infty \), the graph becomes steeper and more negatively inclined, suggesting sharper changes in direction.- As \( k \rightarrow +\infty \), the graph rises and further smooths out, behaving more like a linear function.

Step 6: Connect Discriminant to Graph Shape

The shape of the graph of \( f(x) \) changes primarily due to the location and number of stationary points, which depend on the sign of the discriminant of \( f'(x) \). The number of zeros of \( f'(x) \) influences the curvature and changes in direction in the graph of \( f(x) \).

Key Concepts

Graphing a Cubic Function

Graphing cubic functions helps us visually understand how changes in parameters affect the shape. For the function \( f(x) = x^3 + kx \), graphing it for different values of \( k \) shows dramatic changes. At \( k = 0 \), the graph resembles a typical cubic shape. As \( k \) increases, say to 1 or 2, the graph tilts and shifts, leading to steeper slopes that appear leftward or rightward depending on whether \( k \) is positive or negative. Conversely, decreasing \( k \) below zero results in the graph tilting in the opposite direction. Observing these shifts visually supports the understanding of changes that occur in cubic functions as parameters vary.

Derivatives of Cubic Functions

In calculus, derivatives tell us the slope of a function at any point. For our cubic function \( f(x) = x^3 + kx \), the derivative \( f'(x) \) is found using differentiation rules, resulting in \( f'(x) = 3x^2 + k \). This derivative itself takes the form of a quadratic equation and provides key insights into the function's behavior. The values of \( x \) for which \( f'(x) = 0 \) are the points where the slope of \( f(x) \) changes, indicating possible maxima, minima, or points of inflection. By studying \( f'(x) \), we can predict where and how \( f(x) \) changes, making understanding derivatives a powerful tool when analyzing cubic functions.

Quadratic Discriminant Analysis

The quadratic discriminant indicates the nature of the roots that affect the shape of \( f(x) \). The discriminant of a quadratic function \( ax^2 + bx + c \) is given by \( b^2 - 4ac \). For \( f'(x) = 3x^2 + k \), the discriminant is \( -12k \). At different values of \( k \), this discriminant determines the number and nature of real roots. If \( -12k > 0 \) (i.e., \( k < 0 \)), the discriminant is positive, meaning \( f'(x) \) has two real zeros. This leads to local extrema in \( f(x) \). When the discriminant is zero (\( k = 0 \)), there is one real root, indicating an inflection point. Negative discriminants (\( k > 0 \)) indicate no real roots, meaning \( f'(x) \) has no turning points, resulting in a monotonic \( f(x) \).

Understanding Function Behavior

Analyzing how a function behaves as its parameter values change is crucial. In the case of \( f(x) = x^3 + kx \), varying \( k \) leads to notable shifts in behavior. With \( k = 0 \), the graph has inflection points and is naturally symmetric. When \( k < 0 \), the appearance of both maxima and minima suggests that \( f(x) \) has turning points, highlighting non-linear behavior. If \( k > 0 \), \( f(x) \) becomes monotonic, smoothing out and continuously increasing or decreasing without turning points. Understanding these behaviors assists in predicting and interpreting real-world scenarios modeled by cubic functions.

Monotonic Functions and k's Influence

A function is monotonic if it consistently increases or decreases without changing direction. For the function \( f(x) = x^3 + kx \), the value of \( k \) significantly influences whether the function is monotonic. If \( k > 0 \), the absence of zeros in \( f'(x) = 3x^2 + k \) implies that \( f(x) \) doesn't change its slope direction, maintaining a monotonic behavior either upwards or downwards. When \( k < 0 \), the function changes from monotonic to non-monotonic as local maxima and minima introduce points of inflection. Lastly, as \( k \) approaches either extreme, \( f(x) \) noticeably shifts towards a linear behavior, predicting consistent trends which enhance our understanding of the function's overall nature.