Question

What conclusions can you draw about \(f\) from the information that \(f^{\prime}(c)=f^{\prime \prime}(c)=0\) and \(f^{\prime \prime \prime}(c)>0 ?\)

Short answer

The point \(x = c\) is an inflection point where the function changes from concave down to concave up.

Step-by-step solution

Understanding the Derivatives

We are given that the first derivative of the function, \(f'(c)\), is equal to zero. This suggests that at \(x = c\), the function either has a critical point or an inflection point. Additionally, the second derivative, \(f''(c)\), is also zero, which usually indicates an inflection point. However, since additional information is provided via the third derivative, we need to explore that as well.

Analyzing the Third Derivative

Given \(f'''(c) > 0\), we understand that the third derivative being positive at \(x = c\) suggests the concavity of the function is changing from concave down to concave up. This is key information because it indicates that \(x = c\) is indeed an inflection point, where the function changes its curvature at this point.

Conclude the Nature of \(c\)

Due to \(f'(c) = 0\) and \(f''(c) = 0\), with \(f'''(c) > 0\), the function \(f\) undergoes a change in concavity at \(c\). Thus, \(x = c\) is neither a local maximum nor a local minimum but is an inflection point, as it is a point where the function changes concavity. The positive third derivative indicates that to the left of \(c\), the function is concave down, and to the right, it is concave up.

Key concepts

Critical Point

A critical point of a function is where the first derivative, denoted as \(f'(x)\), is zero or undefined. At such points, the function may have a local maximum, local minimum, or a point of inflection. In simpler terms, this is where the slope of the tangent to the curve is flat.

Understanding critical points is essential because they tell us important information about the behavior of the function. If you imagine a mountain range, the peaks and valleys are like the local maxima and minima, and the places where the slope is exactly flat (but still part of the landscape) are the critical points. However, not every critical point is necessarily a peak or a valley; it can also be where the function changes its curvature direction.

To fully classify a critical point, you must look at higher-order derivatives. Just knowing that \(f'(c) = 0\) is not enough to determine the precise nature of the critical point.

Concavity

Concavity is a concept that tells us which way a function curves. It's determined using the second derivative of the function, \(f''(x)\).

- If \(f''(x) > 0\), the graph is concave up, like holding a cup the right way up, and the function looks like a smile.- If \(f''(x) < 0\), it is concave down, as if the cup is upside down, and the function forms a frown.

An inflection point occurs where the concavity changes. At these points, \(f''(x)\) usually equals zero, and the third derivative comes into play for further analysis. It's like going from climbing uphill to starting to go downhill—there is a subtle change in direction. At an inflection point, although the function is not at a peak or a valley, it reflects a shift from concave up to concave down, or vice versa. Concavity is crucial because it influences how the function's values change, showing the acceleration or deceleration in the rate of change of the function.

Third Derivative

The third derivative, \(f'''(x)\), provides insight into how the concavity is changing at a given point. When someone says \(f'''(c) > 0\), it implies the function's concavity is increasing at \(x = c\). This means there is a change in the direction of the curve's concavity, marking \(x = c\) as an inflection point.

- A positive third derivative suggests that the function is moving from a concave down shape to a concave up shape as you pass through that point. - Conversely, if the third derivative were negative, the function would shift from concave up to concave down.

The role of the third derivative is like being the detailed analyzer of the curvature behavior. Where the second derivative's equal to zero usually indicates a static change point, the third derivative confirms the nature of this change, making sure it's not a flat-line scenario but rather a dynamic twist in the graph's journey from one direction of concavity to another. This allows you to conclude more definitively about what happens at what might initially seem just like any critical or suspicious point on the curve.