Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) is differentiable, has domain \([0,6]\), and has three local maxima and two local minima on \((0,6)\).

Step-by-Step Solution

Answer

The graph of such a function is possible and must be smooth with three maxima and two minima within \((0,6)\).

Step 1: Understand Differentiability

Differentiability means that the function has a well-defined derivative at every point within the domain \([0,6]\). The graph must be smooth without any sharp corners or discontinuities.

Step 2: Analyze the Given Domain

The function is defined on the interval \([0,6]\). The domain includes the endpoints 0 and 6, so the graph must start at \(x = 0\) and finish at \(x = 6\).

Step 3: Analyze Local Maxima and Minima

There should be three local maxima and two local minima within the interval \((0,6)\). Local maxima mean that these points have a peak when compared to nearby points, while local minima mean these points are a trough when compared to nearby points.

Step 4: Sketch the General Shape of the Graph

Start by plotting possible points for three peaks (local maxima) and two troughs (local minima) within \((0,6)\). The graph should rise to a peak, dip to a trough, then repeat this pattern to meet the requirement of three maxima and two minima.

Step 5: Connect the Points Smoothly

Ensure the function is smooth and differentiable. Draw a curve that smoothly connects the points representing the maxima and minima, ensuring no sharp turns or discontinuities.

Step 6: Verify the Graph

Check the graph to confirm it meets all specified conditions: it is defined on \([0,6]\), it has three local maxima and two local minima inside \((0,6)\), and it is smooth and differentiable throughout.

Key Concepts

Local Maxima and Minima

In calculus, identifying local maxima and minima is vital for understanding the behavior of differentiable functions. Local maxima occur at points where a function reaches a peak in its immediate vicinity, meaning the value of the function at this point is greater than at any nearby points. Conversely, local minima occur at points where the function reaches a trough, with the value at this point being lower than surrounding values.

To determine where these points happen, you could use the first derivative test. The points where the derivative changes sign indicate potential maxima or minima. If the derivative changes from positive to negative, it's a local maximum. Change from negative to positive signifies a local minimum.

These critical points help shape the graph, combining with other elements of calculus to describe the entire function within its domain.

Graph Sketching

Graph sketching, especially for functions with specified properties like local maxima or minima, involves careful analysis. Start by identifying the domain—in this case, \[0,6\]. Next, think about the location of the peaks and troughs, as these will mark the positions of local maxima and minima. The function must have three maxima and two minima.

• Use these maxima and minima to plot the fundamental structure of the curvature.
• Ensure the graph has a balanced number of rises and falls, smoothly transitioning from one point to the next.
• Double-check continuity between your plotted points; the graph should not have any breaks or sharp edges.

These initial steps provide a framework to sketch the graph, respecting the given properties of the function while maintaining differentiability.

Calculus

Calculus is a fundamental mathematical tool used to study how functions change. It lays the groundwork for understanding deep concepts like differentiation and integration. Differentiation allows us to find derivatives, giving insights into the rate of change of the function.

• This process helps us locate slopes, tangent lines, and local extrema such as maxima and minima.
• Derivatives are key to applying the first and second derivative tests which confirm the nature of slopes at critical points.

This toolkit not only informs us about static properties but also predicts dynamic behavior across the function's domain. Calculus essentially helps in creating an accurate picture of how a curve behaves.

Continuous Functions

A function is continuous if you can draw its graph without lifting your pencil. In mathematical terms, it means there are no breaks, jumps, or holes in the function. For the function given with domain \[0,6\], it must be defined at every point within this range.

Continuity ensures that between any two points on the graph, all intermediate points are also present on the graph. A continuous function over an interval helps maintain the smoothness required for differentiability.

• Ensure values flow consistently throughout the interval.
• All differentiable functions are continuous, but not all continuous functions are differentiable.

For a smooth graph with local maxima and minima, every point flexibly connects to its neighbors, establishing a seamless function across the entire domain.