Problem 115 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(c\) is a critical number of \(f\) where \(a

Step-by-Step Solution

Answer

The statement is true. Since \(c\) is a critical number in the interval \((a,b)\) and \(f^{\prime\prime}(c)<0\), the Second Derivative Test confirms that \(f\) has a relative maximum at \(x=c\).

Step 1: Analyze the given conditions

We have \(c\) as a critical number of \(f\) with \(a

Step 2: Apply the Second Derivative Test

According to the Second Derivative Test, if \(f'(c)=0\) and \(f''(c)<0\), then \(f\) has a relative maximum at \(x=c\). As we already know from the given conditions, \(f'(c)=0\) and \(f^{\prime\prime}(c)<0\). Thus, we can conclude that the statement is true - If \(c\) is a critical number of \(f\) where \(a

Key Concepts

Critical Number

When analyzing the behavior of a function, identifying the critical numbers can be extremely helpful. A critical number of a function is any number c in the domain of the function where its derivative is zero, f'(c) = 0, or does not exist. Why is this important? Critical numbers are potentially where the function could have a relative maximum or minimum, or a point of inflection.

In our exercise, we're given that c is a critical number within the interval (a, b). The crucial aspect to note here is that the derivative at c is zero which doesn't inherently tell us if it's a maximum, minimum, or neither; this is where further analysis is required and where the second derivative test comes into play.

Relative Maximum

A relative maximum of a function at a point x=c is a point where the function's value is higher than all other values in some interval around c. Think of it as a local peak in the graph of the function. To confirm a relative maximum using calculus, we often turn to the First Derivative Test or the Second Derivative Test.

In the second step of the solution, we applied the Second Derivative Test, which states that if f'(c) = 0 and f''(c) < 0, then the function has a relative maximum at x=c. The negative second derivative indicates that the graph is concave down, creating a 'hilltop' shape, which signifies a relative maximum. This is because when the second derivative is negative, the slope of the function is decreasing, which aligns with the decrease in slope you experience as you reach the top of a hill and start going down.

Concavity

The concept of concavity describes the curvature of a graph. If the graph of a function is concave up, it curves upwards like a smiley face, and any line segment between two points on the graph will lie beneath the graph. Conversely, if the graph is concave down (like a frown), then the line segment will lie above the graph. This characteristic is important when predicting the behavior of function values between different points.

Mathematically, concavity is determined by the sign of the second derivative. If f''(x) > 0, the function is concave up. If f''(x) < 0, the function is concave down. Beyond identifying the shape of the graph, the concavity tells us something about the acceleration of the function's growth or decline. For our problem, a negative second derivative across an interval (a, b), which includes our critical number c, implies that the function is concave down throughout that interval, suggesting a 'downhill' curvature from left to right at c, pointing to a relative maximum at that critical number.

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Problem 114

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Problem 117

Other exercises in this chapter

Problem 113

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the graph o

View solution

Problem 114

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the graph o

View solution

Problem 117

Show that the quadratic function $$ f(x)=a x^{2}+b x+c \quad(a \neq 0) $$ is concave upward if \(a>0\) and concave downward if \(a

View solution

Problem 118

Consider the functions \(f(x)=x^{3}, g(x)=x^{4}\), and \(h(x)=-x^{4}\) a. Show that \(x=0\) is a critical number of each of the functions \(f, g\), and \(h .\)

View solution

Key Concepts

Critical Number

When analyzing the behavior of a function, identifying the critical numbers can be extremely helpful. A critical number of a function is any number c in the domain of the function where its derivative is zero, f'(c) = 0, or does not exist. Why is this important? Critical numbers are potentially where the function could have a relative maximum or minimum, or a point of inflection.

In our exercise, we're given that c is a critical number within the interval (a, b). The crucial aspect to note here is that the derivative at c is zero which doesn't inherently tell us if it's a maximum, minimum, or neither; this is where further analysis is required and where the second derivative test comes into play.

Relative Maximum

A relative maximum of a function at a point x=c is a point where the function's value is higher than all other values in some interval around c. Think of it as a local peak in the graph of the function. To confirm a relative maximum using calculus, we often turn to the First Derivative Test or the Second Derivative Test.

In the second step of the solution, we applied the Second Derivative Test, which states that if f'(c) = 0 and f''(c) < 0, then the function has a relative maximum at x=c. The negative second derivative indicates that the graph is concave down, creating a 'hilltop' shape, which signifies a relative maximum. This is because when the second derivative is negative, the slope of the function is decreasing, which aligns with the decrease in slope you experience as you reach the top of a hill and start going down.

Concavity

The concept of concavity describes the curvature of a graph. If the graph of a function is concave up, it curves upwards like a smiley face, and any line segment between two points on the graph will lie beneath the graph. Conversely, if the graph is concave down (like a frown), then the line segment will lie above the graph. This characteristic is important when predicting the behavior of function values between different points.

Mathematically, concavity is determined by the sign of the second derivative. If f''(x) > 0, the function is concave up. If f''(x) < 0, the function is concave down. Beyond identifying the shape of the graph, the concavity tells us something about the acceleration of the function's growth or decline. For our problem, a negative second derivative across an interval (a, b), which includes our critical number c, implies that the function is concave down throughout that interval, suggesting a 'downhill' curvature from left to right at c, pointing to a relative maximum at that critical number.