Question

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

Short answer

The function \(f(x)\) may have a local maximum, a local minimum, or neither at \(x=c\), although \(f'(c)=0\). This is determined by the behavior of \(f'(x)\) around \(x=c\).

Step-by-step solution

Understanding the Problem

Initially, a given function \(f(x)\) is a differentiable function, and the derivative of the function \(f'(c)=0\) at a point \(c\), which lies within the function's domain. The question is whether there is a local maximum or minimum at \(x=c\). The purpose here is to determine whether the derivative's turning point \(x=c\) will generate a minimum or maximum for function \(f(x)\).

Applying the First Derivative Test

Apply the First Derivative Test to identify potential local maximums and minimums. The First Derivative Test states that if \(f'(x)\) changes sign when passing through \(x=c\), then \(f(x)\) has a local extremum at this point: If the sign changes from positive to negative, \(f(x)\) will have a local maximum at \(x=c\). If the sign changes from negative to positive, then \(f(x)\) has a local minimum at \(x=c\).

Analyze the Result

From the previous step, note that if there is no change in sign for \(f'(x)\) around \(x=c\), then \(f(x)\) does not have a local extremum point at \(x=c\). In summary: When \(f'(c) = 0\), \(f(x)\) could have a local maximum, minimum, or neither at \(x=c\). This depends on the behavior of \(f'(x)\) around \(x=c\).

Key concepts

Local Maximum and Minimum

When studying the features of a curve, identifying the high and low points is a fundamental step. These points, known as the local maximum and minimum, are where a function reaches its greatest or least value within a neighborhood around a given point. It is at these points that the curve changes direction.

Let's say we consider a smooth, continuous hill. The very top of the hill represents a local maximum, because it is the highest point in the immediate area. Conversely, if you stand in a small valley, that point is a local minimum as it is the lowest point around.

Mathematically, if a function's derivative switches from positive to negative, this indicates that the function was increasing before the point and begins to decrease after it—hinting at a local maximum. If the derivative changes from negative to positive, the function was decreasing and then starts increasing, signaling a local minimum. However, these conditions are not guarantees on their own — they are only suggestive of a local max or min, which brings us to the significance of the First Derivative Test.

Differentiable Function

At the heart of calculus, the concept of differentiation plays a crucial role in understanding the behavior of functions. A differentiable function is simply one that can be differentiated at all points in its domain. In practical terms, this means the graph of the function has no sharp corners, breaks, or cusps — it is smooth and has a defined slope at every interior point.

For example, consider a function that describes the smooth path of a car driving along a curvy road. At any given moment, you can determine the direction the car is heading and how steep the path is, which corresponds to calculating the derivative of that function at that point. In addition, the differentiability of a function ensures that the First Derivative Test can be applied to investigate the existence of local maxima or minima.

First Derivative Sign Change

One of the most revealing characteristics of a function comes from observing how the sign of its first derivative changes. A sign change in the derivative is a strong indicator of a function's turning points, which are potential extrema of the function.

Imagine you are tracking the speed of a runner over time. If the runner slows down to a stop (speed becomes zero) and then starts running backwards, there was a sign change in the speed—from positive to negative. Similarly, in calculus, if the first derivative of a function goes from positive to negative at a point, the function is possibly at a local maximum.

Conversely, if the derivative changes from negative to positive, the function may be at a local minimum. It's like the runner stopping and reversing direction from backward to forward. This change can be visualized by examining the slope of the tangent line to the curve of the function, which is what the first derivative represents.

Extrema of a Function

Exploring extrema, which is the collective term for the minimum and maximum values of a function, is a key use for derivatives in calculus. These points represent the peak values that the function can reach, either locally within a confined interval or globally across the entire domain of the function.

Mathematically, identifying the extrema often involves finding points where the first derivative equals zero (critical points) and then using the First Derivative Test to assess how the function behaves around these points. It's the critical step in distinguishing whether we have found a hilltop (local maximum), a valley (local minimum), or perhaps a flat plateau where the function neither rises nor falls (neither a max nor a min).

However, it's vital to remember that not every point where the derivative equals zero is a point of extrema. This highlights the importance of context and additional checks, such as the Second Derivative Test or examining the function over its domain, to conclusively establish the presence of a local or global extremum.