Problem 30 - 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. $$\begin{array}{c|c|c|c|r|r|r|r|r|r|r|c}\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline f^{\prime}(x) & 5 & 2 & 1 & -2 & -5 & -3 & -1 & 2 & 3 & 1 & -1 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Critical points: near \(x = 3\) (local max), near \(x = 7\) (local min), near \(x = 10\) (local max).

Step 1: Identify Critical Points by Finding Where f' Changes Sign

Critical points occur where the derivative changes from positive to negative or negative to positive. Looking at the table:- From \(x = 2\) to \(x = 3\), the sign changes from \(1\) to \(-2\) implying a critical point near \(x = 2\) or \(x = 3\).- From \(x = 5\) to \(x = 6\), the sign changes from \(-3\) to \(-1\) implying no sign change here.- From \(x = 6\) to \(x = 7\), the sign changes from \(-1\) to \(2\) implying a critical point near \(x = 6\) or \(x = 7\).- From \(x = 9\) to \(x = 10\), the sign changes from \(1\) to \(-1\) implying a critical point near \(x = 9\) or \(x = 10\).

Step 2: Determine the Nature of Each Critical Point

To determine if a critical point is a maximum, minimum, or neither, observe the sign change:- At \(x = 2\) or \(x = 3\), \(f'\) changes from positive to negative, indicating a local maximum.- At \(x = 6\) or \(x = 7\), \(f'\) changes from negative to positive, indicating a local minimum.- At \(x = 9\) or \(x = 10\), \(f'\) changes from positive to negative, indicating a local maximum.

Key Concepts

Derivative

The concept of a derivative is foundational to calculus and serves as a tool to understand the behavior of functions. In simple terms, the derivative of a function at a point provides the rate at which the function's value changes at that point. For example, if you imagine driving a car, the derivative is similar to the car's speedometer, telling you how fast, or slow, the function value is changing.

To compute the derivative, we differentiate the function, which involves calculating \(f'(x)\), often through rules like the power rule, product rule, or chain rule. The derivative can vary over different segments of x, changing from positive to negative, or vice-versa. Understanding these changes in sign helps in identifying critical points, which are key points where the derivative is zero or undefined. These points may reveal important characteristics of the function, such as peaks, valleys, or flat segments.

Sign Change

Sign changes in derivatives are crucial for finding critical points. Whenever the derivative of a function changes its sign, it indicates a possible critical point. A critical point is where the function can have a maximum or minimum value.

Sign changes are analyzed by observing the derivative's transition from positive to negative, or from negative to positive:

• If \(f'(x)\) changes from positive to negative, it suggests the function was increasing and then begins to decrease, hinting a local maximum.
• If \(f'(x)\) changes from negative to positive, it implies the function was decreasing and then starts to increase, indicating a local minimum.

This understanding aids in predicting the behavior of the function graph at those points. For example, in the provided exercise, the sign of the derivative changes at specific points like from \(x = 2\) to \(x = 3\) or from \(x = 9\) to \(x = 10\), signaling potential turning points.

Local Maximum

A local maximum is a point where the function reaches its highest value relative to nearby points. This is like climbing a hill and reaching the top, knowing that going forward or backward will both lead you downhill.

The identification of a local maximum involves examining the sign change of the derivative as follows:

• A local maximum occurs where \(f'(x)\) changes from positive to negative.

In a function's graph, at a local maximum, the curve reaches a peak point before descending. During problem-solving, you might see a change, like from \(f'(x) = 2\) to \(f'(x) = -2\), marking such instances. These points are important as they help us find where the greatest value of the function appears within a given interval. In the exercise, at \(x = 2\) or \(x = 3\), \(f'(x)\) changed from positive to negative, confirming a local maximum.

Local Minimum

A local minimum is just the opposite of a local maximum. It's a point where a function achieves the lowest value when compared to its immediate vicinity.

Imagine hiking in a valley, surrounded by taller peaks on either side. You've reached a local minimum.

The sign change pattern for identifying a local minimum is:

• A local minimum occurs where \(f'(x)\) changes from negative to positive.

On a graph, the curve dips downward into a trough and then rises again on either side of a local minimum. As seen in the exercise, this happens for \(f'(x)\) when moving from a negative to a positive value, like from \(f'(x) = -1\) to \(f'(x) = 2\). Such changes pinpoint valleys in the function, offering insight into where it reaches its lowest points.