Problem 17 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 17 through 20, sketch the graph of a function \(f\) that has all the given properties. a. \(f^{\prime}(x)>0\) when \(x<0\) and when \(x>5\) b. \(f^{\prime}(x)<0\) when \(00\) when \(-62\) d. \(f^{\prime \prime}(x)<0\) when \(x<-6\) and when \(-3

Step-by-Step Solution

Answer

The function increases to the left of 0 and right of 5, and decreases between 0 and 5. Concave up on intervals (-6, -3) and (2, ∞), and concave down on intervals (-∞, -6) and (-3, 2).

Step 1: Understand the first derivative conditions

The given information tells us about the behavior of the first derivative, which indicates the slope of the function. When the first derivative, \(f^{\backprime}(x) > 0\), the function is increasing. When \(f^{\backprime}(x) < 0\), the function is decreasing.- For \(x < 0\) and \(x > 5\), \(f^{\backprime}(x) > 0\), so the function is increasing in these intervals. - For \(0 < x < 5\), \(f^{\backprime}(x) < 0\), so the function is decreasing in this interval.

Step 2: Understand the second derivative conditions

The second derivative indicates the concavity of the function. When the second derivative \(f^{\backprime\backprime}(x) > 0\), the function is concave up (shaped like a U). When \(f^{\backprime\backprime}(x) < 0\), the function is concave down (shaped like an upside-down U).- For \(-6 < x < -3\) and \(x > 2\), \(f^{\backprime\backprime}(x) > 0\), so the function is concave up in these intervals.- For \(x < -6\) and \(-3 < x < 2\), \(f^{\backprime\backprime}(x) < 0\), so the function is concave down in these intervals.

Step 3: Plotting the graph

Use the information about increasing/decreasing and concavity to sketch the graph:- For \(x < -6\), the function is concave down. This means it will look like an upside-down U. - For \(-6 < x < -3\), the function is concave up. This means it will look like a U.- For \(-3 < x < 0\), the function is concave down and increasing before reaching 0.- For \(0 < x < 2\), the function is concave down and decreasing in this interval.- For \(2 < x < 5\), the function is concave up and still decreasing.- For \(x > 5\), the function is concave up and increasing.Sketch a rough graph ensuring these behaviors and changes in concavity occur at the given intervals.

Key Concepts

First Derivative

The first derivative, denoted as \(f'(x)\), informs us about the slope of the function \(f(x)\). In simpler terms, it tells us if the function is going up or down at any specific point.
When \(f'(x) > 0\), the slope is positive, meaning the function is increasing. Conversely, when \(f'(x) < 0\), the slope is negative, indicating the function is decreasing. Here’s a breakdown for this specific exercise:

• For \(x < 0\) and \(x > 5\), \(f'(x) > 0\), so the function is increasing in these intervals.
• For \(0 < x < 5\), \(f'(x) < 0\), so the function is decreasing in this interval.

By studying the behavior of the first derivative, you can understand where the function is rising or falling.

Second Derivative

To grasp the concept of the second derivative, denoted as \(f''(x)\), think of it as the curvature or the shape of our function. While the first derivative tells us the slope, the second derivative tells us how 'curvy' the graph is.
When \(f''(x) > 0\), the function is concave up, resembling a U shape. When \(f''(x) < 0\), the function is concave down, like an upside-down U.
For this exercise:

• For \(-6 < x < -3\) and \(x > 2\), \(f''(x) > 0\), so the function is concave up in these intervals.
• For \(x < -6\) and \(-3 < x < 2\), \(f''(x) < 0\), so the function is concave down in these intervals.

Understanding the second derivative helps in sketching the graph's shape, giving you an idea of where the graph forms U or inverted U shapes.

Concavity

Concavity describes how the curve bends. Whether it bends upwards or downwards is determined by the second derivative.
To determine concavity:

• If \(f''(x) > 0\), the function \(f(x)\) bends upwards and is concave up.
• If \(f''(x) < 0\), the function \(f(x)\) bends downwards and is concave down.

In this exercise:

• The function is concave down for \(x < -6\) and \(-3 < x < 2\).
• It is concave up for \(-6 < x < -3\) and \(x > 2\).

Identifying concavity is crucial as it impacts how the graph of the function will look in these intervals.

Increasing and Decreasing Functions

The concepts of increasing and decreasing functions are fundamental in understanding the overall behavior of the graph. They are closely tied to the first derivative \(f'(x)\).

• An increasing function means that as \(x\) moves to the right, \(f(x)\) goes up. This happens when \(f'(x) > 0\).
• A decreasing function means that as \(x\) moves to the right, \(f(x)\) goes down. This happens when \(f'(x) < 0\).

This exercise shows:

• The function increases on intervals \(x < 0\) and \(x > 5\).
• The function decreases on the interval \(0 < x < 5\).

Recognizing these regions helps in plotting the slopes of the function accurately.