Problem 21 - Applied Calculus

Question

Suppose \(f\) has a continuous derivative whose values are given in the following table. (a) Estimate the \(x\) -coordinates of critical points of \(f\) for \(0 \leq x \leq 10 .\) (b) For each critical point, indicate if it is a local maximum of \(f\), local minimum, or neither. $$ \begin{array}{l|l|l|l|r|r|r|r|l|l|l|c} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline f^{\prime}(x) & 5 & 2 & 1 & -2 & -5 & -3 & -1 & 2 & 3 & 1 & -1 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Critical points near x=2.5, 6.5, and 9.5; local max at 2.5 and 9.5, local min at 6.5.

Step 1: Identify Critical Points

Critical points occur where the derivative is zero or changes sign. Examine the values of \(f'(x)\) provided in the table to identify changes in sign or where \(f'(x) = 0\).- From \(x = 2\) to \(x = 3\), \(f'(x)\) changes from 1 to -2, indicating a critical point between 2 and 3.- From \(x = 5\) to \(x = 6\), \(f'(x)\) changes from -3 to -1; both are negative, but there's a zero between these points.- From \(x = 6\) to \(x = 7\), \(f'(x)\) changes from -1 to 2, indicating a critical point between 6 and 7.- From \(x = 9\) to \(x = 10\), \(f'(x)\) changes from 1 to -1, indicating a critical point between 9 and 10.

Step 2: Determine the Nature of Each Critical Point

To determine whether each critical point is a local maximum, minimum, or neither, analyze the sign changes of \(f'(x)\):- For the critical point between 2 and 3, \(f'(x)\) changes from positive to negative, indicating a local maximum.- For the critical point between 5 and 6, \(f'(x)\) has no sign change (both values are negative), so neither a maximum nor a minimum is indicated here.- For the critical point between 6 and 7, \(f'(x)\) changes from negative to positive, indicating a local minimum.- For the critical point between 9 and 10, \(f'(x)\) changes from positive to negative, indicating a local maximum.

Key Concepts

Continuous Derivative

A continuous derivative is an essential concept in calculus that ensures the function behaves predictably over its domain. If a function has a continuous derivative, it means the function's rate of change doesn't have jumps or gaps. This property allows us to find critical points by analyzing where the derivative equals zero or changes sign. Critical points provide valuable insights into the behavior of the function.

For instance, in the problem we're examining, knowing that the derivative is continuous within the interval \(0 \leq x \leq 10\), allows us to predict the existence of critical points just by observing sign changes in the derivative's values.

Continuous derivatives are often visualized as smooth curves without sharp corners, with the derivative providing a consistent picture of how the function is changing at each point.

• Ensures predictability of function behavior
• Smooth transitions in the function's graph
• Facilitates analysis of critical points

Understanding continuous derivatives forms a foundation for exploring how functions behave on intervals and how to use derivatives to optimize functions or locate extrema.

Local Maximum

A local maximum occurs at a critical point where the function reaches a higher value than at any other point nearby. To identify a local maximum, we typically analyze the sign change of the derivative. If the derivative changes from positive to negative at a critical point, it indicates a local maximum. This occurs because the function is increasing before the critical point and decreasing after it.

In our problem, at the critical point between \(x = 2\) and \(x = 3\), the derivative changes from 1 to -2. This change from positive to negative signals a local maximum. At this point, the function reaches a peak before descending again. Recognizing local maximums is critical as they can indicate the peak performance or optimal conditions within certain contexts.

• Defined by a change in \( f'(x) \) from positive to negative
• Indicates a peak or turning point
• Useful in optimization and analysis

Grasping the concept of local maximum helps in understanding and predicting where functions reach their highest values locally.

Local Minimum

A local minimum is found at a point where the function's value is lower than that at nearby points, often described as a valley. To determine if a critical point is a local minimum, we need to look at the derivative's behavior: if it changes from negative to positive, a local minimum is likely present.

In the exercise provided, between \(x = 6\) and \(x = 7\), the derivative switches from -1 to 2. This change reveals a local minimum, as the function transitions from decreasing to increasing. Identifying local minimum points is valuable for understanding where functions bottom out, crucial for minimizing cost or loss in practical scenarios.

• Recognized by a \( f'(x) \) sign change from negative to positive
• Represents valleys or troughs in the function
• Important for minimizing outcomes

With a firm grasp on the idea of local minimum, one can better analyze regions where the function achieves its lowest points locally, often an objective in various applications.

Sign Change Analysis

Sign change analysis is a method used to determine where a function's derivative changes sign, indicating potential critical points. By closely examining these changes, we can better identify where the function may reach local maximums or minimums.

The process involves noting where in our interval the derivative goes from positive to negative, suggesting a local maximum, or from negative to positive, suggesting a local minimum. For the given problem, sign change analysis helps us find critical points between \(x = 2\) and \(x = 3\), between \(x = 6\) and \(x = 7\), and between \(x = 9\) and \(x = 10\).

• Helps identify critical points by analyzing \( f'(x) \)
• Indicates potential extrema (maxima and minima)
• Aids in graph interpretation and function behavior analysis

By applying sign change analysis, we simplify the task of pinpointing where functions reach their peaks and valleys, a fundamental skill for tackling calculus problems efficiently.