Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

Point \(c\) is an inflection point with concavity changing upwards.

Step 1: Understand the Given Information

We start by examining the conditions given in the problem: \( f^{\prime}(c)=0 \) implies that \( c \) is a critical point of the function \( f \). This is where the function may have a local maximum, minimum, or a point of inflection. \( f^{\prime\prime}(c)=0 \) suggests that there is neither a local maximum nor a minimum, as the second derivative test is inconclusive.

Step 2: Explore the Third Derivative Condition

The condition \( f^{\prime\prime\prime}(c)>0 \) tells us more about the nature of \( c \). The positive third derivative indicates that the function is transitioning or changing its concavity from concave down to concave up at the point \( c \). This suggests a possible point of inflection at \( c \).

Step 3: Draw Conclusion

Based on the above conditions, since \( f^{\prime}(c)=f^{\prime\prime}(c)=0 \) and \( f^{\prime\prime\prime}(c)>0 \), we can conclude that \( c \) is a point of inflection where the concavity of the function is changing, and the graph is moving upward as it passes through \( c \).

Key Concepts

Second Derivative Test

The second derivative test is a useful method to determine the concavity of a function and identify points of local maxima or minima. To use this test, you first find the critical points where the first derivative of the function is zero, i.e., where \( f'(x) = 0 \). These points are candidates for local extreme values.
Once you have a critical point, you evaluate the second derivative \( f''(x) \) at that point. Here's how to interpret the results:

• If \( f''(x) > 0 \), the function is concave up at \( x \), indicating a local minimum.
• If \( f''(x) < 0 \), the function is concave down at \( x \), indicating a local maximum.
• If \( f''(x) = 0 \), the test is inconclusive, and you cannot determine if there is a local maximum or minimum at that point.

In this particular exercise, the second derivative \( f''(c) = 0 \), so the second derivative test doesn't provide information about a maximum or minimum at \( c \). The analysis, therefore, moves to subsequent tests, like the third derivative test.

Third Derivative Test

When both the first derivative \( f'(c) \) and the second derivative \( f''(c) \) are zero, the second derivative test doesn't provide conclusive results. In such scenarios, the third derivative test comes into play to deliver deeper insight. The third derivative \( f'''(c) \) suggests how the function might change around the critical point.
The third derivative test involves:

• If \( f'''(c) > 0 \), there is a change in concavity from concave down to concave up, indicating an inflection point at \( c \).
• If \( f'''(c) < 0 \), the concavity shifts from concave up to concave down, still suggesting an inflection point but with different directional curvature.

In this situation, with \( f'''(c) > 0 \), we conclude there is an inflection point at \( c \) and the graph transitions from a downward to an upward curve. The third derivative gives us valuable information on the nature of the function's change beyond what the first and second derivatives can provide.

Concavity and Inflection Points

Understanding concavity and inflection points is crucial for analyzing the behavior of a function. Concavity describes how the function bends or curves. A function is concave up when it resembles a cup (\( f''(x) > 0 \)), and concave down when it appears like a cap (\( f''(x) < 0 \)).
An inflection point occurs where a function changes its concavity. This means the second derivative shifts signs:

• From positive to negative or vice versa, signaling that the curve changes direction in terms of its bending.
• Inflection points are crucial because they mark transitions in the slope and curvature of the graph.

In the exercise provided, the function at point \( c \) suggests an inflection point because the third derivative is positive \( f'''(c) > 0 \). It indicates a transition where the curve switches from being concave down to concave up. Recognizing such points helps in graphing and understanding the overall behavior of complicated functions.