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, we need to identify where the second derivative changes sign. Start by finding the first derivative of the function: \( f'(x) = 3x^2 - 36x - 10 \). Now, find the second derivative: \( f''(x) = 6x - 36 \).

Step 2: Set the Second Derivative to Zero

Inflection points occur where the second derivative is zero or undefined. Set \( f''(x) = 6x - 36 \) equal to zero: \( 6x - 36 = 0 \).

Step 3: Solve for x

Solve the equation \( 6x - 36 = 0 \). Begin by adding 36 to both sides: \( 6x = 36 \). Then, divide both sides by 6 to isolate \( x \): \( x = 6 \).

Step 4: Confirm the Change of Concavity

To confirm the inflection point, check whether the concavity changes around \( x = 6 \). Choose test points on either side of 6, like 5 and 7. Calculate \( f''(5) = 6 \times 5 - 36 = -6 \), which is negative, indicating concave down. Calculate \( f''(7) = 6 \times 7 - 36 = 6 \), which is positive, indicating concave up. The concavity changes, confirming an inflection point at \( x = 6 \).

Step 5: Find the y-coordinate of the Inflection Point

Substitute \( x = 6 \) back into \( f(x) \) to find the corresponding \( y \)-value: \( f(6) = 6^3 - 18 \times 6^2 - 10 \times 6 + 6 \). Calculate this value: \( 216 - 648 - 60 + 6 = -486 \). The inflection point is at \((6, -486)\).

Step 6: Verify with a Graph

Use a calculator or graphing software to plot \( f(x) = x^3 - 18x^2 - 10x + 6 \). Confirm visually that the graph changes from concave down to concave up at \( x = 6 \), matching the algebraic solution.

Key Concepts

Second Derivative

When dealing with functions, especially in calculus, the second derivative helps us understand how the rate of change of a function's slope varies. It's the derivative of the derivative, essentially giving us information on acceleration or deceleration in different contexts. For a given function \( f(x) \), the second derivative is denoted as \( f''(x) \). In the exercise, we first calculated the second derivative of the function \( f(x) = x^3 - 18x^2 - 10x + 6 \). By differentiating the first derivative, \( f'(x) = 3x^2 - 36x - 10 \), we derived \( f''(x) = 6x - 36 \). This second derivative indicates where we should look for inflection points, as these are points where the concavity might change. Changing concavity, indicated by a sign change in \( f''(x) \), suggests a possible inflection point. Therefore, when the second derivative equals zero or becomes undefined, it can signify a potential change in the concavity, prompting further exploration of the x-values where this occurs.

Concavity

Understanding concavity is crucial in determining the shape and behavior of a graph of a function. If a graph is said to be concave up, it resembles an upward-opening bowl or a smile, and \( f''(x) > 0 \). Conversely, a graph that is concave down resembles an upside-down bowl or a frown, where \( f''(x) < 0 \).An inflection point is the location on the graph where the concavity transitions between up and down. To identify whether the concavity changes around a specific point, like \( x = 6 \) in our problem, we calculate the second derivative at values slightly less than and greater than this point. By plugging in test values, we see a shift: from \( f''(5) = -6 \) indicating concave down, to \( f''(7) = 6 \), showing concave up. This switch confirms an inflection point, supported by different concavity on either side.Recognizing such changes allows mathematicians to deduce critical elements of a function's behavior, guiding the prediction and modeling of real-world phenomena using polynomial functions.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables raised to whole number powers, multiplied by coefficients. They are powerful tools for modeling continuous data and other real-life situations, primarily due to their simple arithmetic operations and the diverse shapes their graphs can produce.In our exercise, we explored a cubic polynomial, \( f(x) = x^3 - 18x^2 - 10x + 6 \). This specific type falls under cubic polynomials due to the highest power of 3. Such functions can have up to three real roots and typically display complex behavior such as inflection points, peaks, and troughs. The degree and coefficients impact these attributes deeply, dictating both the function's range and concavity shifts.Graphing these functions helps visualize their behavior and often confirms algebraic findings, like our identified inflection point at \((6, -486)\). Polynomial equations can model a broad spectrum of scenarios by accommodating various degrees and coefficients, making them indispensable in calculus and broader mathematical analysis.