Problem 39 - Calculus A New Horizon

Question

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval; then use those graphs to estimate the \(x\) -coordinates of the inflection points of \(f\), the intervals on which \(f\) is concave up or down, and the intervals on which \(f\) is increasing or decreasing. Check your estimates by graphing \(f\). $$f(x)=x^{4}-24 x^{2}+12 x, \quad-5 \leq x \leq 5$$

Step-by-Step Solution

Answer

Inflection points at \(x = \pm 2\). Increasing for \((-\infty, -\sqrt{2}) \cup (\sqrt{2}, \infty)\), decreasing otherwise. Concave up for \((-\infty, -2) \cup (2, \infty)\), concave down otherwise.

Step 1: Identify Derivatives

Start by finding the first derivative \(f'\) of \(f(x) = x^4 - 24x^2 + 12x\). Using basic differentiation rules:\[ f'(x) = 4x^3 - 48x + 12 \]Next, find the second derivative:\[ f''(x) = 12x^2 - 48 \].

Step 2: Graph First and Second Derivatives

Use a graphing tool to plot \(f'(x) = 4x^3 - 48x + 12\) and \(f''(x) = 12x^2 - 48\) over the interval \(-5 \leq x \leq 5\). The graph of \(f'(x)\) will help identify where \(f\) is increasing or decreasing, while \(f''(x)\) will help identify concavity and inflection points.

Step 3: Analyze the First Derivative

Examine the graph of \(f'(x)\) to determine where \(f'\) is positive (\(f\) is increasing) and where \(f'\) is negative (\(f\) is decreasing). Note the points where \(f'(x) = 0\) as these could indicate local maxima, minima, or points where the function changes direction.

Step 4: Analyze the Second Derivative

Look at the graph of \(f''(x)\) to identify inflection points, which occur where \(f''(x) = 0\) and the concavity changes. Determine the intervals where \(f''(x) > 0\) (\(f\) is concave up) and \(f''(x) < 0\) (\(f\) is concave down).

Step 5: Verify with the Graph of Function

Graph the original function \(f(x) = x^4 - 24x^2 + 12x\) and compare your findings with the graph. Identify the x-coordinates of inflection points and verify the intervals of concave up/down and increasing/decreasing behavior initially estimated from the derivatives.

Key Concepts

Derivatives

Derivatives are fundamental in calculus, serving as tools to describe how a function changes. For a given function like \(f(x) = x^4 - 24x^2 + 12x\), the first derivative, \(f'(x)\), represents the rate of change or the slope of the function at any point. To find \(f'(x)\), you apply basic differentiation rules:

• For \(x^4\), the derivative is \(4x^3\).
• For \(-24x^2\), it is \(-48x\).
• For \(12x\), the derivative is \(12\).

Thus, the first derivative is \(f'(x) = 4x^3 - 48x + 12\). The second derivative, \(f''(x)\), which is found by differentiating \(f'(x)\), provides information about the curvature of the function. For our function, this results in \(f''(x) = 12x^2 - 48\). This second derivative plays a crucial role in understanding concavity and identifying inflection points.

Concavity

Concavity describes the direction in which a graph curves. The second derivative, \(f''(x)\), is key to determining this. If \(f''(x) > 0\), the function \(f(x)\) is concave up, resembling a U-shape. Conversely, if \(f''(x) < 0\), the function is concave down, resembling an upside-down U.By examining the second derivative \(f''(x) = 12x^2 - 48\), you can identify the intervals of concavity:

• The function is concave up when \(12x^2 - 48 > 0\). Solving this inequality finds where \(x^2 > 4\), which are intervals \(x < -2\) and \(x > 2\).
• The function is concave down in the interval \(-2 < x < 2\), where \(12x^2 - 48 < 0\).

Recognizing these intervals helps sketch the graph's behavior more accurately.

Inflection Points

Inflection points occur where the function changes concavity, going from concave up to concave down or vice versa. These points happen where the second derivative equals zero, as this indicates a potential change in the curve's direction.For \(f''(x) = 12x^2 - 48\), set the equation to zero to find potential inflection points:\[12x^2 - 48 = 0\]Solving gives \(x = \pm 2\). These x-values need to be checked against the changes in sign of \(f''(x)\) to confirm them as inflection points. If the sign of \(f''(x)\) changes as you pass through these x-values, they are indeed points of inflection on the graph of \(f(x)\). This insight refines your understanding of how and where the function might twist in its path.

Increasing/Decreasing Functions

To determine where a function is increasing or decreasing, analyze the sign of its first derivative, \(f'(x)\). If \(f'(x) > 0\), the function is increasing, indicating the function's slope is positive. Conversely, if \(f'(x) < 0\), the function is decreasing, with a negative slope.Examining \(f'(x) = 4x^3 - 48x + 12\), you aim to find where the derivative is positive or negative. Set \(f'(x) = 0\) to help locate potential turning points where the function might switch from increasing to decreasing or vice versa:\[4x^3 - 48x + 12 = 0\]Solving this will provide critical points which, along with a sign chart or graph analysis, allow you to outline:

• The intervals of \(f(x)\) being increasing, evident between the critical points where \(f'(x) > 0\).
• The intervals of decreasing behavior, where \(f'(x) < 0\).

Understanding these concepts thoroughly enhances predictions on the function's behavior and its graph.