Problem 5 - Applied Calculus

Question

(a) Graph a function with two local minima and one local maximum. (b) Graph a function with two critical points. One of these critical points should be a local minimum, and the other should be neither a local maximum nor a local minimum.

Step-by-Step Solution

Answer

(a) A W-shaped curve. (b) A cubic-like curve with an inflection point.

Step 1: Understand the Exercise

The exercise asks for graphical representations. In part (a), we need a function with two dips (local minima) and one rise (local maximum). In part (b), we want two critical points: one should be the lowest point locally (local minimum), and the other should neither rise nor dip (saddle point or inflection).

Step 2: Prepare for Part (a)

For two local minima and one local maximum, think of a graph like a W-shape, where the peaks and troughs are not symmetrical. This will create the required local minima and a single local maximum.

Step 3: Sketch Part (a)

Draw a smooth W-shaped curve. Start from above the x-axis, dip down to a point, rise to a peak, dip to another point lower than or equal to the first, and rise again. Label the dips as local minima and the peak as the local maximum.

Step 4: Prepare for Part (b)

For this function, consider a cubic function which typically has inflection points. Let one point be a local minimum and the other be neither (just a change in concavity).

Step 5: Sketch Part (b)

Draw a cubic-like curve. Start by gently rising, reaching the first local minimum, then climb past a subtle point of inflection (where the graph flattens but doesn't rise or fall sharply). Label these points according to their characteristics.

Key Concepts

Local Minima

In the realm of calculus, a local minimum is a point on a graph where the function reaches a level lower than all nearby points. Imagine walking in a valley; the local minimum would be the lowest part of that valley. This is where the graph dips, creating a trough. Mathematically, if you have a function \(f(x)\), then at a local minimum at \(x = a\), the value \(f(a)\) is less than \(f(x)\) for values close to \(a\).

To identify a local minimum:

• First, compute the derivative \(f'(x)\) and set it to zero, solving \(f'(x) = 0\) to find critical points.
• Next, use the second derivative test. If \(f''(a) > 0\), then \(a\) is a local minimum.

Local minima provide essential insights into the behavior of functions and help to understand graph shapes.

Local Maximum

A local maximum is the opposite of a local minimum. It is where the function reaches a level higher than all nearby points, reminiscent of the peak of a hill in your path. Let's say the function \(f(x)\) has a local maximum at \(x = b\); this means \(f(b)\) is greater than \(f(x)\) for values close to \(b\).

To determine a local maximum:

• Find points where the derivative \(f'(x)\) equals zero, just like with local minima.
• Apply the second derivative test. If \(f''(b) < 0\), this confirms a local maximum.

These peaks tell us where the function's output starts and stops increasing. Local maxima are a crucial part of understanding function behavior and optimizing problems.

Inflection Points

Inflection points mark where a function changes its concavity. These points can be seen as the shoulders on a graph, where it shifts from curving upwards to downwards, or vice versa. Such shifts can make the graph appear to slightly "flatten" out for a moment.

To find inflection points:

• Look for points where the second derivative \(f''(x)\) changes sign. It's essential that the sign genuinely changes rather than remaining constant or not existing.

An example is a cubic function, such as \(y = x^3\), which often shows an inflection point balancing between the curves. At the inflection point, the behavior of the graph transforms, affecting how steeply the function rises or falls.

Graphs of Functions

At its core, graphing functions involves plotting points that show the relationship between variables, typically \(x\) and \(y\) on a 2D plane. Understanding how functions behave visually is vital. Graphs not only depict local minima and maxima, but also illustrate inflection points, providing a story of the function’s rate of change.

To effectively graph functions:

• Identify key features like intercepts, which are the points where the graph crosses the axes.
• Pinpoint and label critical points—local minima and maxima, as well as inflection points.
• Acknowledge the end behavior, observing what happens as \(x\) approaches infinity or negative infinity.

Such analyses enable better comprehension of mathematical models, allowing us to anticipate and calculate important parameters in various fields from physics to economics.