Problem 10 - Applied Calculus

Question

For \(f(x)=x^{3}-18 x^{2}-10 x+6,\) find the inflection point algebraically. Graph the function with a calculator or computer and confirm your answer.

Step-by-Step Solution

Answer

The inflection point is at \((6, -486)\).

Step 1: Find the Second Derivative

To find the inflection point of the function \(f(x)=x^3-18x^2-10x+6\), we first need to calculate its second derivative. Begin by finding the first derivative:\[f'(x) = 3x^2 - 36x - 10.\] Next, find the second derivative, which is the derivative of the first derivative:\[f''(x) = 6x - 36.\]

Step 2: Set the Second Derivative to Zero

To find the inflection point, set the second derivative \(f''(x)\) equal to zero and solve for \(x\):\[6x - 36 = 0.\] Solve the equation for \(x\):\[6x = 36,\] which simplifies to \[x = 6.\]

Step 3: Verify the Concavity Change

To ensure that \(x = 6\) is indeed an inflection point, the concavity of the function must change at this point. Test values on either side of \(x = 6\) in the second derivative \(f''(x) = 6x - 36\). For \(x = 5\), \(f''(5) = 6(5) - 36 = -6\), which is negative, indicating concave down. For \(x = 7\), \(f''(7) = 6(7) - 36 = 6\), which is positive, indicating concave up. Since the concavity changes from concave down to concave up, \(x = 6\) is an inflection point.

Step 4: Find the Coordinates of the Inflection Point

Calculate the corresponding \(y\)-coordinate by substituting \(x = 6\) into the original function \(f(x)\):\[f(6) = (6)^3 - 18(6)^2 - 10(6) + 6.\] Simplify:\[f(6) = 216 - 648 - 60 + 6 = -486.\] Thus, the inflection point is \((6, -486)\).

Step 5: Confirm with a Graph

Graph the function \(f(x) = x^3 - 18x^2 - 10x + 6\). The graph should show the curve changing concavity at the point \((6, -486)\), confirming that this is indeed the inflection point.

Key Concepts

Second Derivative

The second derivative of a function gives us insightful information about the concavity and potential inflection points. Essentially, when you're exploring a function's graph, this derivative helps to determine how the function's slope is changing.

• The first derivative, denoted as \( f'(x) \), represents the slope or rate of change of the function. It's the measure of how steep the graph is at any given point.
• The second derivative, denoted as \( f''(x) \), subsequently gives us the rate of change of this slope, or the concavity of the function.

Finding the second derivative involves taking the derivative of the first derivative. In the exercise, the first derivative \( f'(x) = 3x^2 - 36x - 10 \) was derived from the original function \( f(x)=x^3-18x^2-10x+6 \). Then, differentiating again gives \( f''(x) = 6x - 36 \). This equation serves to find points of inflection where the nature of the curvature changes.

Concavity Change

A change in concavity occurs at an inflection point. It signifies the transition of a graph from being concave upwards to concave downwards, or vice versa. To determine these transitions, we look at the second derivative:

• If \( f''(x) > 0 \), the function is concave up, resembling a cup or a smile.
• If \( f''(x) < 0 \), it's concave down, similar to a frown.

In the exercise, solving \( f''(x) = 6x - 36 = 0 \) indicated \( x = 6 \) as a potential point of inflection. However, confirming that \( x = 6 \) is a true inflection point requires checking if the concavity changes from positive to negative or vice versa around this value. Testing numbers like \( x = 5 \) and \( x = 7 \), you observe a concavity change from down to up, validating \( x = 6 \) as the inflection point. This signifies a real shift in the graph's curvature at this point.

Graphing Functions

Graphing functions visually confirms the mathematical analysis of derivatives and concavity. When graphing a function, you're essentially plotting all points \( (x, y) \), where \( y = f(x) \), to visualize its behavior.

• Inflection points can be identified more easily on a graph as the place where the curve changes its "bending" direction.
• By plotting the given function, \( f(x) = x^3 - 18x^2 - 10x + 6 \), one can observe the concavity changing at the confirmed coordinate \( (6, -486) \).

Utilizing graphing calculators or software can greatly facilitate the process of visual identification. Through graphing, alongside computational methods, confirmation is achieved for analytical findings in the exercise. Hence, visual tools combined with calculus are powerful in understanding complex behaviors of functions.