Problem 33 - Calculus A New Horizon

Question

In each part, assume that \(a\) is a constant and find the inflection points, if any. (a) \(f(x)=(x-a)^{3}\) (b) \(f(x)=(x-a)^{4}\)

Step-by-Step Solution

Answer

(a) Inflection point at \(x = a\). (b) No inflection point.

Step 1: Differentiate the function for (a)

To find the inflection points of the function \(f(x) = (x-a)^3\), we first calculate the second derivative. Start by finding the first derivative: \( f'(x) = 3(x-a)^2 \).

Step 2: Differentiate again for (a)

Now, find the second derivative of \(f(x) = (x-a)^3\): \( f''(x) = 6(x-a) \).

Step 3: Find inflection points for (a)

To find the inflection points, set the second derivative equal to zero and solve for \(x\): \( 6(x-a) = 0 \), which simplifies to \( x = a \). This means there is an inflection point at \(x = a\).

Step 4: Differentiate the function for (b)

For the function \(f(x) = (x-a)^4\), compute the first derivative: \( f'(x) = 4(x-a)^3 \).

Step 5: Differentiate again for (b)

Now find the second derivative of \(f(x) = (x-a)^4\): \( f''(x) = 12(x-a)^2 \).

Step 6: Analyze inflection point conditions for (b)

To find inflection points, set the second derivative equal to zero and solve for \(x\): \( 12(x-a)^2 = 0 \). The only solution is \( x = a \). However, since the concavity does not change (as the second derivative is always non-negative), there is no inflection point.

Key Concepts

Understanding the Second Derivative

The second derivative of a function is a powerful tool in calculus. It provides valuable insights into the behavior of the function's graph, particularly with regard to its curvature or concavity. By taking the derivative of a derivative, you essentially assess how the rate of change is changing. In our original exercise, we calculated the second derivatives for the functions

• For part (a): we found the second derivative of \(f(x) = (x-a)^3\) to be \(f''(x) = 6(x-a)\).
• For part (b): the second derivative of \(f(x) = (x-a)^4\) came out to be \(f''(x) = 12(x-a)^2\).

Calculating the second derivative is integral to identifying concavity and potential inflection points.

Discovering Concavity with Derivatives

Concavity describes how a function curves around its critical points. The second derivative helps establish whether a function is concave up or down. - A positive second derivative (\(f''(x) > 0\)) indicates the function is concave up, like a cup. - A negative second derivative (\(f''(x) < 0\)) means it is concave down, like a frown. When examining \(f(x) = (x-a)^4\), we realized that the second derivative \(f''(x) = 12(x-a)^2\) never becomes negative. This informs us that \(f(x)\) is always concave up, and hence, the concavity never changes. These findings are crucial in determining potential inflection points since we look for changes in concavity at these points.

The Role of Calculus in Finding Inflection Points

Calculus provides the framework for understanding changes within a function through the use of derivatives. An inflection point is where a function changes its concavity, crucial for understanding the function's overall shape. To locate inflection points, you need to:

• Find the second derivative.
• Set it equal to zero and solve for \(x\).

For \(f(x) = (x-a)^3\), the second derivative \(f''(x) = 6(x-a)\) gives us \(x = a\). This solution implies a potential change in concavity, indicating an inflection point at \(x = a\). In contrast, for \(f(x) = (x-a)^4\), while \(x = a\) solves the equation \(f''(x) = 0\), the graph's concavity does not shift, thus excluding the existence of an inflection point at this solution. Understanding these principles with the help of calculus ensures precision in identifying and analyzing inflection points effectively.