Problem 66 - Essential Calculus

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of \(f(x)=1 / x\) is concave downward for \(x<0\) and concave upward for \(x>0\), and thus it has a point of inflection at \(x=0\)

Step-by-Step Solution

Answer

The statement is false. The function \( f(x) = 1 / x \) is concave upward for \( x < 0 \) and concave downward for \( x > 0 \). Moreover, the function is not defined at \( x = 0 \), so it cannot have a point of inflection at \( x = 0 \).

Step 1: Understand the Concepts

Concavity refers to the 'bending' of the function. A function is concave up on an interval if the function lies above the tangent on that interval. Conversely, a function is concave down on an interval if the function lies below the tangent on that interval. An inflection point is a point on the graph of the function where the concavity changes.

Step 2: Derive the Function

To find the inflection point, one must first find the second derivative of the function. The derivative of \( f(x) = 1 / x \) is \( f'(x) = -1 / x^2 \). The second derivative is then \( f''(x) = 2 / x^3 \).

Step 3: Determine the Concavity

The sign of the second derivative tells us the concavity. If \( f''(x) > 0 \), then the function is concave up. If \( f''(x) < 0 \), the function is concave down. For \( x < 0 \), \( f''(x) > 0 \), hence, the function is concave up. For \( x > 0 \), \( f''(x) < 0 \), hence, the function is concave down.

Step 4: Identify the Point of Inflection

An inflection point is a point where the concavity changes. However, in this case, the function \( f(x) = 1/x \) is not defined at \( x = 0 \), so there can't be a point of inflection at \( x = 0 \).

Key Concepts

Concavity of a Function

When studying the shape of a graph, understanding the concavity is crucial. Concavity describes the curvature direction of a function's graph. Imagine holding a graph from either side and bending it; the direction it curves into is a reflection of its concavity.

A function is considered to be concave upward (or concave down) if, loosely speaking, it curves up (down), resembling the shape of a smile (frown). Mathematically, if you were to draw a line tangent to any point on this section of the curve, the function's graph would sit above (below) the line for concave up (down).

Visualizing Concavity

Picture each segment of a roller coaster. Where the ride pulls you upwards, the curve of the track is akin to a function's concave up section. When the coaster dives downward, it resonates with a function that is concave down. This curvature is not just about aesthetics; it has implications for the behavior of the function, especially where economics and physics equations come into play, dictating trends and points of equilibrium.

Moreover, real-world examples like a dish's shape can illustrate concavity—concave up mimics a bowl; concave down represents an upside-down bowl.

Second Derivative Test

The second derivative test is an efficient way to determine a function’s concavity and consequently the nature of its stationary points (where the first derivative is zero). The essence of this test lies in the sign of the second derivative at a particular point.

Using the Second Derivative Test

If you find that the second derivative, denoted as f''(x), is positive at a certain point, the function exhibits concave upward behavior there. Conversely, a negative second derivative signifies that the function is concave downward at that point. This information not only reveals the concavity but also can indicate whether a stationary point is a local minimum (concave up) or maximum (concave down).

• If f''(x) > 0: The function is concave up and x is a local minimum.
• If f''(x) < 0: The function is concave down and x is a local maximum.

However, should the second derivative equal zero, the test becomes inconclusive, and you might need to rely on other methods to make assertions about the function's behavior at that point.

Inflection Points in Calculus

Inflection points are like the plot twists in the story of a function’s graph. These intriguing points mark where the function changes concavity. Where the function switches from a concave upward to a concave downward or vice-versa, you'll find an inflection point.

Detecting Inflection Points

Inflection points are typically found by examining where the second derivative changes signs. It’s like following the trail of breadcrumbs that the second derivative lays down; a change in sign is a signal that you’ve stumbled upon a potential inflection point.

However, there's a catch: a change in the second derivative’s sign is a necessary, but not sufficient, condition for an inflection point. The point of potential inflection must exist on the function’s domain. That is why the function f(x) = 1/x, cannot have an inflection point at x = 0, because the function is not defined there; there's no value of f(x) when x is zero. Think of it as looking for a treasure on a map; if the 'X' marks a spot outside the map’s edges, there’s simply no treasure to be found.