Problem 1 - Calculus Early Transcendentals

Question

In each part, sketch the graph of a continuous function \(f\) with the stated properties. (a) \(f\) is concave up on the interval \((-\infty,+\infty)\) and has exactly one relative extremum. (b) \(f\) is concave up on the interval \((-\infty,+\infty)\) and has no relative extrema. (c) The function \(f\) has exactly two relative extrema on the interval \((-\infty,+\infty),\) and \(f(x) \rightarrow+\infty\) as \(x \rightarrow+\infty\) (d) The function \(f\) has exactly two relative extrema on the interval \((-\infty,+\infty),\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow+\infty\)

Step-by-Step Solution

Answer

Sketch parabolas and cubics as per concavity and relative extrema.

Step 1: Understand Conavity and Extrema

Before sketching any graph, let's define the terms: A function is concave up on an interval if its second derivative is positive over that interval. Relative extrema refer to local maximum or minimum points on the graph. Now, let's address each part of the exercise individually.

Step 2: Sketch for Part (a)

For a function that is concave up everywhere and has exactly one relative extremum, we can think of a simple quadratic function. A good example is the function \( f(x) = x^2 \). It is concave up on \(( -\infty, +\infty )\) and has one relative minimum at \( x = 0 \). In the graph sketch, draw a parabola opening upwards with its vertex at \( (0, 0) \).

Step 3: Sketch for Part (b)

Now consider a function that is concave up everywhere but has no relative extrema. A common example is an exponential function like \( f(x) = e^x \). This function is increasing everywhere, has no turns, and remains concave up since its second derivative is positive. The curve rises exponentially as \( x \) increases. Draw this function starting from near zero at very negative \( x \), rising to the right.

Step 4: Sketch for Part (c)

For a function with two relative extrema and \( f(x) \rightarrow +\infty \) as \( x \rightarrow +\infty \), imagine a cubic polynomial like \( f(x) = x^3 - 3x \). This function has turning points, hence relative extrema, and increases to infinity as \( x \rightarrow +\infty \). Sketch the cubic curve with a "W" shape, ensuring it passes in an upward direction at both extremes.

Step 5: Sketch for Part (d)

For a function with two relative extrema and \( f(x) \rightarrow -\infty \) as \( x \rightarrow +\infty \), modify the cubic from part (c) by negating it: \( f(x) = -x^3 + 3x \). This function will have a similar "M" shape, starting from infinity as \( x \rightarrow -\infty \) and going to \(-\infty\) as \( x \rightarrow +\infty \). Draw the cubic curve starting high on the left and ending low on the right.

Key Concepts

Concavity

In mathematics, the concept of concavity is important in understanding the shape of a curve on a graph. A function is said to be

• Concave up if its second derivative is positive over an interval.
• Concave down if its second derivative is negative.

When a function is concave up, it resembles the shape of a bowl or a smile. This means that the slope of the function's tangent lines keeps increasing as you move along the curve. One intuitive way to think about concavity is to imagine how the curve hugs the horizontal axis.

For example, the function \( f(x) = x^2 \) is concave up across the entirety of the real number line \((-\infty, +\infty)\). Its second derivative, \( f''(x) = 2 \), is always positive, indicating the curvature is consistently upward. Similarly, exponential functions like \( f(x) = e^x \) are also concave up since the slope increases as \( x \) grows.

Relative Extrema

Relative extrema are the local maxima or minima of a function. They are critical for identifying peaks or valleys on a graph. These points are found where the first derivative of a function equals zero or is undefined, indicating a potential turning point.

To determine whether it is a maximum or minimum, you analyze the first and second derivatives:

• A local maximum occurs at a critical point if the function changes from increasing to decreasing, i.e., a sign change from a positive to a negative derivative.
• A local minimum occurs if the function changes from decreasing to increasing, indicated by a sign change from a negative to a positive derivative.

For instance, in the function \( f(x) = x^3 - 3x \), relative extrema exist where the derivative \( f'(x) = 3x^2 - 3 \) is zero, yielding critical points. Evaluating further with the second derivative \( f''(x) = 6x \) helps determine the nature of these extrema dependent on the concavity at these points.

Infinite Intervals

Infinite intervals refer to the domain in which a function continuously stretches towards positive or negative infinity. These intervals are crucial for understanding the ultimate behavior of functions as \(x\) moves towards extreme values, whether in the positive or negative direction.

Considering functions over infinite intervals helps us predict if they will continue to rise or fall indefinitely. For example:

• A polynomial such as \( f(x) = x^3 - 3x \) behaves differently at each end of its interval. Here, as \( x \rightarrow +\infty \), the function \( f(x) \) approaches \(+\infty\).
• Conversely, modifying it to \( f(x) = -x^3 + 3x \) changes the behavior, such that as \( x \rightarrow +\infty \), \( f(x) \rightarrow -\infty \).

Understanding these intervals assists in sketching and predicting the long-term behavior of graphs to ensure proper function analysis.