Problem 33 - Calculus with Concepts in Calculus

Question

Plot the graph of \(f^{\prime}\), and then use the Newton-Raphson method to approximate all values of \(c\) for which \((c, f(c))\) is an inflection point. Continue until the output of the calculator does not change. $$ f(x)=x^{7}-x^{5}+x^{3}+3 x^{2} $$

Step-by-Step Solution

Answer

Inflection points at \(c \approx 0\).

Step 1: Find the First Derivative

Calculate the first derivative of the function \( f(x) = x^{7} - x^{5} + x^{3} + 3x^{2} \). Use the power rule for differentiation to find: \( f'(x) = 7x^{6} - 5x^{4} + 3x^{2} + 6x \).

Step 2: Find the Second Derivative

Calculate the second derivative, which is necessary to find inflection points. Differentiate \( f'(x) \) with respect to \( x \): \( f''(x) = 42x^{5} - 20x^{3} + 6x + 6 \).

Step 3: Set the Second Derivative to Zero

To find potential inflection points, solve \( f''(x) = 0 \). This yields the equation: \( 42x^{5} - 20x^{3} + 6x + 6 = 0 \).

Step 4: Use Newton-Raphson Method

Apply the Newton-Raphson method to approximate the roots of \( f''(x) = 0 \). The Newton-Raphson formula is \( x_{n+1} = x_n - \frac{f''(x_n)}{f'''(x_n)} \). First, compute \( f'''(x) = 210x^{4} - 60x^{2} + 6 \).

Step 5: Iterate Until Convergence

Choose initial guesses close to where inflection points might occur, based on inspection of the graph or symmetry arguments. Use \( x_0 = 0 \) and iterate using \( x_{n+1} = x_n - \frac{42x_n^{5} - 20x_n^{3} + 6x_n + 6}{210x_n^{4} - 60x_n^{2} + 6} \). Continue until the solution converges (two or more iterations yield the same result).

Step 6: Validate Inflection Points

Check the change in the sign of the second derivative around the computed values to ensure they are indeed inflection points.

Key Concepts

Inflection Points

Inflection points of a function are where the concavity changes. This means the curve shifts from concave up (looking like a cup) to concave down (looking like a cap), or vice versa. At these points, the second derivative of a function is equal to zero.

• Not all points where the second derivative is zero are inflection points. There must be a sign change in the second derivative around the point for it to be classified as an inflection point.
• Identifying these points is important in understanding the behavior and shape of the graph of a function.

To find the inflection points, we must first find the first and second derivatives of the function. Then, set the second derivative to zero and solve for the values of the variable. These values, after verification of a sign change, will give us the inflection points.

First Derivative

The first derivative of a function gives us the slope of the tangent line at any point on the function. It's crucial in determining where a function is increasing or decreasing.

• To find the first derivative, we apply rules of differentiation like the power rule. For instance, if you have a term like 10x^4, its derivative would be 40x^3.
• A positive first derivative means the function is increasing, while a negative first derivative indicates that the function is decreasing.

In the given problem, we find the first derivative of the function using these rules: \( f'(x) = 7x^6 - 5x^4 + 3x^2 + 6x \).This expression tells us where the slope of the function changes, helping identify potential maxima and minima.

Second Derivative

The second derivative of a function tells us about the concavity of the function. It's useful in finding inflection points by revealing where concavity changes.

• If the second derivative is positive, the function is concave up, resembling a "U" shape.
• If it's negative, the function is concave down, resembling an upside-down "U".
• A zero value in the second derivative signals a potential inflection point, although a sign change is required to confirm it.

For our function \( f(x) \), the second derivative is calculated as \( f''(x) = 42x^5 - 20x^3 + 6x + 6 \). Solving \( f''(x) = 0 \) helps us locate potential inflection points by solving for \( x \) values where this derivative zeroes out.

Differentiation

Differentiation is a mathematical process used to find the derivative of a function. This process allows us to understand the rate of change of a function at any given point.

• The basic rules of differentiation include the power rule, product rule, and chain rule, among others.
• For polynomial functions, like the one in the exercise, the power rule is typically employed. To differentiate a term \( ax^n \), you bring down the exponent \( n \) and multiply it by \( a \), then reduce the exponent by 1 to get \( a \, n \, x^{n-1} \).

In the context of the problem, differentiation allows us to find both the first and second derivatives. These derivatives provide insights into the slope and concavity of the function, key to determining inflection points with accuracy.