Problem 6 - Calculus Early Transcendentals

Question

Sketch a function that changes from concave up to concave down as \(x\) increases. Describe how the second derivative of this function changes.

Step-by-Step Solution

Answer

Short answer: The function \(f(x) = x^3 - 3x^2\) changes from concave up to concave down as \(x\) increases. The function is concave up when \(x < 1\) and concave down when \(x > 1\), with an inflection point at \(x = 1\).

Step 1: Choose a function

Let's start with a popular function that exhibits a change in concavity: the cubic function \(f(x) = x^3 - 3x^2\).

Step 2: Find the first and second derivative of the function

Find the first derivative of the function, \(f'(x)\), which represents the function's slope: $$ f'(x) = \frac{d}{dx}(x^3 - 3x^2) = 3x^2 - 6x. $$ Next, find the second derivative, \(f''(x)\), which represents the curvature or concavity of the function: $$ f''(x) = \frac{d}{dx}(3x^2 - 6x) = 6x - 6. $$

Step 3: Determine the change of concavity

To find the point at which the function changes from concave up to concave down, we examine when \(f''(x)\) transitions from positive to negative. We start by setting the second derivative equal to zero: $$ 6x - 6 = 0 \Rightarrow x = 1. $$ When \(x < 1\), the second derivative \(f''(x) > 0\) hence the function is concave up. When \(x > 1\), the second derivative \(f''(x) < 0\) hence the function is concave down.

Step 4: Sketch the function

Now, we can sketch the function \(f(x) = x^3 - 3x^2\) and its change in concavity. We note that the function is concave up for \(x < 1\) and concave down for \(x > 1\). Make sure to include these behaviors when sketching the graph. The point where the change in concavity occurs is the inflection point, which is at \(x = 1\).

Key Concepts

Second Derivative Test

Understanding the concavity of a graph is crucial in calculus, and the second derivative test is a powerful tool for this purpose. The test revolves around the second derivative of a function, represented as \( f''(x) \). When applying the test, we search for critical points where \( f'(x) = 0 \) and then evaluate the second derivative at those points.

If \( f''(x) > 0 \) at a critical point, the function's graph is concave up at that point, which typically indicates a local minimum. Conversely, if \( f''(x) < 0 \) at a critical point, the function's graph is concave down, suggesting a local maximum. If the second derivative equals zero or does not exist, the test is inconclusive, and other methods must be used to determine the function's behavior at that point. Let's consider a cubic function, \(f(x) = x^3 - 3x^2\), as an example. Upon calculating its second derivative, \(f''(x) = 6x - 6\), we find a critical point at \(x = 1\). Substituting this back into the second derivative, we observe that the value transitions from positive to negative, signifying a change in concavity.

Concave Up and Down

Graphic representations of functions can exhibit different shapes, and the terms 'concave up' and 'concave down' describe these shapes. A function's graph is concave up when it resembles the shape of a smile or a cup, meaning it opens upwards. Mathematically, this is when the second derivative \( f''(x) \) is positive, indicating that the slope of the tangent line is increasing.

On the other hand, a concave down graph looks like a frown or an upside-down cup. In this situation, the second derivative is negative, and the slope of the tangent line is decreasing. Using our example function \(f(x) = x^3 - 3x^2\), you can visualize the function as concave up for \(x < 1\) where \(f''(x) > 0\), and concave down for \(x > 1\) as \(f''(x) < 0\). The concavity affects many aspects of a function's behavior, including optimization and motion analysis.

Inflection Point

An inflection point is where a function's graph changes concavity—from concave up to concave down or vice versa. It represents a point of the function where the curvature changes sign. Detecting an inflection point involves finding where the second derivative changes sign, which usually occurs at points where \( f''(x) = 0 \) or where the second derivative does not exist.

For the function \(f(x) = x^3 - 3x^2\), the inflection point occurs at \(x = 1\), where the second derivative \(f''(x) = 0\). At this point, the graph of the function moves from being concave up (for \(x < 1\)) to being concave down (for \(x > 1\)). Inflection points are crucial in determining the most detailed possible sketch of a graph, as they mark the transition of curvature and aid in understanding the function's overall geometry.