Question

Writing to Learn If \(f(x)\) is a twice-differentiable function and \(f^{\prime \prime}(c)=0\) at an interior point \(c\) of \(f^{\prime}\) s domain, must \(f\) have an inflection point at \(x=c ?\) Explain.

Short answer

No, the function \(f\) does not necessarily have an inflection point at \(x=c\) even if \(f^{\prime \prime}(c)=0\). An inflection point requires a change in the concavity of the function, which does not follow directly from the second derivative being zero at point \(c\).

Step-by-step solution

Understanding inflection point

A point \(a\) in the domain of function \(f\) is an inflection point if and only if \(f\) is continuous at \(a\), and the function changes its concavity at \(a\). This means that the function changes from being concave up to concave down or vice versa. It is important to note that, to determine the concavity of a function, we usually look at the sign of its second derivative.

Evaluating the given condition

According to the problem, the second derivative of function \(f\) at \(c\) is zero, i.e. \(f^{\prime \prime}(c)=0 \). This means that at the point \(x=c\), the rate of change of slope (which is given by the second derivative) is zero.

Inflection point determination

Even though \(f^{\prime \prime}(c)=0\), it does not necessarily mean \(f\) has an inflection point at \(x=c\). To be an inflection point, the function should change concavity at that point. This requires that the sign of the second derivative changes around the point \(c\), but since the derivative equals zero, we cannot conclude directly whether a change of concavity indeed happens.

Key concepts

Second Derivative

The second derivative of a function, often denoted as f''(x), is a mathematical tool that tells us about the rate at which the function's slope is changing. If you imagine a car's travel being represented on a graph by a function, the first derivative, f'(x), would represent the car's speed, and the second derivative would indicate the car's acceleration.

In the context of the graph of the function, the sign of the second derivative at a particular point reveals whether the graph is concave up or concave down at that point. Concave up is represented by a positive second derivative, meaning the slope is increasing as you move from left to right, akin to the upward curve of a smile. Conversely, concave down is indicated by a negative second derivative, with the slope decreasing from left to right, mimicking the downturn of a frown.

Concavity of a Function

Concavity refers to the curvature of a function's graph. To visually understand this concept, picture a function's curve on a graph. Where the curve is shaped like the interior of a bowl or cup, it is said to be 'concave up.' Alternatively, when the graph forms an arch-like shape, similar to an upside-down bowl, it is 'concave down.'

The concavity of a function is determined by the second derivative, where the disk of the second derivative test comes into play. For an interval where f''(x) is positive, the function exhibits concave up behavior, while a negative f''(x) results in a concave down curve. Identifying the concavity helps in understanding the behavior of the function and predicting its graph's shape. An inflection point, a particular interest area, is where the concavity switches from up to down or vice versa. Nevertheless, simply having a second derivative of zero at a point doesn't guarantee an inflection point; the concavity must change on either side of that point.

Rate of Change

The term 'rate of change' in mathematics broadly describes how one quantity changes in relation to another. For functions that graph the relationship between two variables, the rate of change is an essential concept in calculus and facilitates an understanding of how swiftly values of the function are increasing or decreasing as the input changes.

The first derivative of a function, f'(x), indicates the instantaneous rate of change of the function relative to the input variable 'x.' It represents the slope or gradient at any given point on the curve. Building on this, the second derivative, f''(x), informs us about the rate at which this slope itself is changing, which can highlight regions of acceleration or deceleration in the rate of change. This acceleration or deceleration can be crucial when evaluating a function's behavior and predicting how it will proceed based on current patterns.