Problem 43 - Calculus Single Variable

Question

In Exercises \(43-48,\) the second derivative \(f^{\prime \prime}\) of a function \(f\) is given. Determine every \(x\) at which \(f\) has a point of inflection. $$ f^{\prime \prime}(x)=x(x+1) $$

Step-by-Step Solution

Answer

Points of inflection are at \( x = -1 \) and \( x = 0 \).

Step 1: Understand the Second Derivative

To determine points of inflection, we start by analyzing the behavior of the second derivative given as \( f''(x) = x(x+1) \). A point of inflection occurs where the second derivative changes sign.

Step 2: Find Where the Second Derivative Equals Zero

Set the second derivative equal to zero to find the critical points: \( x(x+1) = 0 \). This gives us \( x = 0 \) and \( x = -1 \) as potential points of inflection.

Step 3: Test the Intervals Around Critical Points

To confirm a change in sign of \( f''(x) \), evaluate the second derivative in the intervals between and around these points. Check intervals: \((-\infty, -1)\), \((-1, 0)\), and \((0, \infty)\).

Step 4: Evaluate the Signs of the Second Derivative

For \( x < -1 \), choose \( x = -2 \): \( f''(-2) = -2(-2+1) = 2 \) (positive). For \( -1 < x < 0 \), choose \( x = -0.5 \): \( f''(-0.5) = -0.5(0.5) = -0.25 \) (negative). For \( x > 0 \), choose \( x = 1 \): \( f''(1) = 1(2) = 2 \) (positive).

Step 5: Identify Points of Inflection from Sign Changes

The sign of \( f''(x) \) changes at \( x = -1 \) (from positive to negative) and at \( x = 0 \) (from negative to positive), confirming that these are points of inflection.

Key Concepts

Second Derivative

The second derivative of a function provides valuable insights into the curvature of its graph. In calculus, when we say second derivative, we refer to the derivative of the derivative. It is denoted as \( f''(x) \). This mathematical tool helps in understanding how the rate of change itself is changing.

In simpler terms:

• The first derivative, \( f'(x) \), shows the rate of change of the function, indicating the slope of the tangent at any point.
• The second derivative, \( f''(x) \), indicates the concavity or curvature of the function's graph.

When \( f''(x) > 0 \), the function is concave up, resembling a ``U'' shape. Conversely, when \( f''(x) < 0 \), the function is concave down, similar to an upside-down "U." This curvature knowledge is crucial for determining points of inflection.

Points of Inflection

A point of inflection is a point on the graph of a function where the concavity changes. For a point to be classified as an inflection point, the function must switch from being concave up to concave down, or vice versa.

This change can only happen at points where the second derivative equals zero or is undefined. However, just because the second derivative is zero does not guarantee a point of inflection. You must confirm a sign change in \( f''(x) \) around these points. Consider the function in our exercise:

• Given \( f''(x) = x(x+1) \), set \( f''(x) = 0 \).
• This provides potential inflection points at \( x = 0 \) and \( x = -1 \).

To confirm, evaluate \( f''(x) \) on intervals around these points to check for a sign change.

Sign Change

The sign change of the second derivative is crucial in solidifying points of inflection. Once potential inflection points are identified by setting \( f''(x) = 0 \), the next step is to analyze intervals around these points to observe if the sign of \( f''(x) \) switches.

Here's how to perform a sign test:

• Choose test points from intervals around the critical points.
• For the function \( f''(x) = x(x+1) \):
• Test \( x = -2 \) for the interval \((-1, -1)\), resulting in a positive value.
• Test \( x = -0.5 \) for \((-1, 0)\), yielding a negative value.
• Test \( x = 1 \) for \((0,   )\), yielding a positive value.

The sign switching at \( x = -1 \) from positive to negative and at \( x = 0 \) from negative to positive confirms these as inflection points.

Critical Points

Critical points hold importance in calculus as they represent where the derivative of a function is zero or undefined, leading to potential maximas, minimas, or inflection points. However, for points of inflection specifically, we focus on where the second derivative, \( f''(x) \), equals zero.

In relation to our exercise:

• We identified \( x = 0 \) and \( x = -1 \) as critical points from \( f''(x) = x(x+1) = 0 \).
• These critical points lay where the trade-off in concavity occurs.

A critical point becomes a point of inflection only when there is a sign change in \( f''(x) \) around it. Distinguishing between simply having a critical point and a true point of inflection is essential in calculus. This ensures an accurate understanding of the graph's behavior at these points.