Question

Interpreting the derivative The graph of \(f^{\prime}\) on the interval [-3,2] is shown in the figure. a. On what interval(s) is \(f\) increasing? Decreasing? b. Find the critical points of \(f .\) Which critical points correspond to local maxima? Local minima? Neither? c. At what point(s) does \(f\) have an inflection point? d. On what interval(s) is \(f\) concave up? Concave down? e. Sketch the graph of \(f^{\prime \prime}\) f. Sketch one possible graph of \(f\)

Short answer

Question: Determine the intervals of increase and decrease, the critical points, the inflection points, and the intervals of concavity, and sketch the graph of the second derivative and an example of the original function using the given graph of the first derivative, \(f'(x)\). Answer: To answer this question, analyze the graph of the derivative, \(f'(x)\), to find the intervals of increase and decrease. Then, determine the critical points and classify them as local maxima, local minima, or saddle points. Next, identify the inflection points and intervals of concavity by examining the slope of the first derivative. Finally, use this information to sketch the graph of the second derivative, \(f''(x)\), and an example of the original function, \(f(x)\).

Step-by-step solution

Increasing and decreasing intervals

To find the intervals where \(f\) is increasing and decreasing, we need to analyze the graph of \(f'\). The function will be increasing when \(f'(x) > 0\) and decreasing when \(f'(x) < 0\). Observe the graph of \(f'\) on the interval \([-3, 2]\) and find the intervals where the graph is above and below the x-axis.

Critical points and maxima/minima

The critical points occur when \(f'(x) = 0\) or when \(f'(x)\) is undefined. Again, we need to analyze the graph of \(f'\), particularly at the points where it intersects the x-axis. Determine the x-values of these points and classify them as local maxima, local minima, or saddle points by analyzing the change in the sign of \(f'(x)\) (e.g., from positive to negative or vice versa).

Inflection points and intervals of concavity

Inflection points occur when the second derivative of the function changes its sign. To find the inflection points, we need to analyze the first derivative (\(f'(x)\)) for changes in the slope. The slope of \(f'(x)\) represents the second derivative (\(f''(x)\)). Inflection points occur when the slope increases or decreases through zero. Identify the points on the graph of \(f'(x)\) where this occurs. After finding the inflection points, we can also identify the intervals where \(f\) is concave up or concave down.

Sketch the graph of \(f''\)

Based on the information we found about the second derivative, we can sketch an approximate graph of \(f''(x)\). Identify the points where \(f''(x) = 0\), i.e., the points of inflection, and the intervals where \(f''(x)\) is positive or negative. Sketch the graph accordingly.

Sketch a possible graph of \(f(x)\)

Using all the information we found, sketch an example of the graph of the original function \(f(x)\). Include crucial points like local maxima, local minima, and inflection points from the previous steps. Consider intervals of increase, decrease, concavity, and the inflection points to shape the graph accurately. Remember that this is just one of the many possible graphs that could correspond to the given derivative.

Key concepts

Increasing and Decreasing Intervals

To determine where a function is increasing or decreasing, we look at the first derivative, which is often noted as \(f'(x)\). If \(f'(x) > 0\), this means that the function \(f(x)\) is increasing on that interval. Conversely, if \(f'(x) < 0\), then \(f(x)\) is decreasing.

So, when analyzing the graph of \(f'(x)\):

• Check when the curve is above the x-axis (positive values). These are the increasing intervals.
• Look for where the curve dips below the x-axis (negative values). These are the decreasing intervals.

For students, a tip is to imagine the graph behavior like a hill: going uphill means increase, and downhill decrease. Analyze the graph within any given interval to decide these behaviors accurately.

Critical Points

Critical points of a function occur where its first derivative \(f'(x)\) is zero or undefined. These points can be local minima, local maxima, or neither (often related to turning points or saddle points).

Steps to find and analyze critical points include:

• Identifying x-values where \(f'(x) = 0\) or \(f'(x)\) is not defined.
• Checking the sign changes of \(f'(x)\) around these points.
• If \(f'(x)\) changes from positive to negative, it indicates a local maximum.
• If \(f'(x)\) changes from negative to positive, it indicates a local minimum.

Consider these points as places where the function could "turn" in its path of the graph. Not every critical point is necessarily a peak or valley, so further analysis of the derivative's behavior is often needed.

Inflection Points

Inflection points occur where a function's concavity changes. This can be spotted by examining the first derivative's graph \(f'(x)\) since it relates to the second derivative.

An inflection point is where the second derivative, \(f''(x)\), changes sign.

• A positive to negative change in \(f''(x)\) signifies a change from concaving "up" to "down."
• A negative to positive change signifies a switch from concaving "down" to "up."

When analyzing \(f'(x)\), focus on the slope:

• Inflection points occur where this slope of \(f'(x)\) moves through zero, shifting from increasing to decreasing, or vice versa.

These points are often where the graph has a noticeable change in direction, like a bend on a graph line where it moves from curving one way to the other.

Concavity

Concavity describes how a function curves, essentially how it opens. Concave up means the graph looks like a smiley face (U shape), while concave down looks like a frown (∩ shape). Checking concavity helps understand the nature of the function between critical points.

Steps to understand concavity include:

• Use \(f''(x)\) to decide the intervals of concavity.
• When \(f''(x) > 0\), the function is concave up in that interval.
• When \(f''(x) < 0\), the function is concave down.

Concavity gives insight into how the function behaves across different intervals, especially its steepness and direction changes. By evaluating concavity, students can predict and sketch more accurate graphs for a given function, integrating in maximum and minimum points seen from \(f'(x)\).