Problem 82 - 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 \(f^{\prime \prime}(x)<0\) on \((a, b)\) and \(f^{\prime}(c)=0\) where \(a

Step-by-Step Solution

Answer

The statement is true. If \(f''(x)<0\) on \((a, b)\) and \(f'(c) = 0\) where \(a

Step 1: Analyze the given information

We're given the following: 1. \(f''(x)<0\) on \((a, b)\): This means that the function is *concave down* on the interval \((a, b)\). In other words, the curvature of the graph is downwards. 2. \(f'(c) = 0\) where \(a < c < b\): This means that at \(x=c\), the slope of the tangent is 0. This could indicate a local maximum or local minimum.

Step 2: Identify the properties of concave down functions

In a function that is *concave down*, a critical point where the first derivative is equal to zero will be a *local maximum*. Since \(f''(x) < 0\) and \(f'(c) = 0\), we can conclude that \(f(c)\) is a local maximum.

Step 3: Determine if the local maximum is also the absolute maximum

Now, we need to check if that local maximum is also the absolute maximum in the closed interval \([a, b]\). In a function that is concave down, if we have a local maximum at some point \(c\) within the interval \((a, b)\), then moving from \(x=c\) to either endpoint \(x=a\) or \(x=b\) will give us lower function values of \(f(a)\) and \(f(b)\) respectively. Thus, by the properties of a concave down function, if there exists a local maximum at \(x=c \in (a, b)\), it will be the absolute maximum of the function in the closed interval \([a, b]\).

Step 4: Conclusion

The statement is true. If \(f''(x)<0\) on \((a, b)\) and \(f'(c) = 0\) where \(a

Key Concepts

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), plays a crucial role in determining the concavity of a function. When \( f''(x) < 0 \), it means the function is concave down in the interval where this condition holds. This is depicted graphically as the graph of the function curving downwards.

The concavity provided by the second derivative informs us about the behavior of a function's slope: as we move from left to right on a interval where the function is concave down, the slope of the tangent line (which is the first derivative) decreases. Thus, the second derivative test is a valuable tool to determine if a critical point is a local maximum or minimum; a negative second derivative at a critical point is an indicator of a local maximum.

However, the second derivative alone cannot tell us if that local maximum is also the absolute maximum. For that, we need to evaluate the function's values throughout the entire interval, especially at the endpoints, to make the distinction.

Critical Points

Critical points of a function occur where the first derivative is either zero or undefined. These points are important because they are potential locations of local extremum (maximums or minimums) and inflection points.

To find a critical point, one would set the first derivative of the function equal to zero and solve for \( x \). However, not every critical point signifies a local extremum. For example, critical points can occur at the inflection points of a function where the concavity changes but the function does not achieve a local maximum or minimum.

Determining whether a critical point corresponds to a local maximum or minimum requires further analysis, often involving the second derivative test or the first derivative test where the sign of the slope changes are examined around the critical point.

Local Maximum

A local maximum of a function is a point where the function value is greater than all other function values in its immediate vicinity. At a local maximum point \( c \), the first derivative of the function \( f'(c) \) will be zero because the slope of the tangent to the curve at that point is horizontal.

When the function is concave down around that point, as indicated by a negative second derivative, that flat tangent represents the highest local value of the function, hence a local maximum. The existence of a local maximum does not necessarily imply the highest value in the given domain—this is where the concept of absolute maximum comes into play, comparing all function values within the domain.

Absolute Maximum

An absolute maximum is the highest function value over the entire domain. Unlike a local maximum, which is only concerned with values in a neighborhood around a point, the absolute maximum is the supreme value in the entire interval being considered.

In a closed interval \([a, b]\), we must evaluate the function at the critical points and at the interval's endpoints to determine the absolute maximum. For functions that are concave down throughout this interval, a local maximum found within the interval will be greater than or equal to the function values at the endpoints; hence, it will also be the absolute maximum.

In problems where we know the endpoints and the concavity between them, finding the function values at the endpoints and any critical points allows us to identify the absolute maximum with certainty.

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Key Concepts

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), plays a crucial role in determining the concavity of a function. When \( f''(x) < 0 \), it means the function is concave down in the interval where this condition holds. This is depicted graphically as the graph of the function curving downwards.

The concavity provided by the second derivative informs us about the behavior of a function's slope: as we move from left to right on a interval where the function is concave down, the slope of the tangent line (which is the first derivative) decreases. Thus, the second derivative test is a valuable tool to determine if a critical point is a local maximum or minimum; a negative second derivative at a critical point is an indicator of a local maximum.

However, the second derivative alone cannot tell us if that local maximum is also the absolute maximum. For that, we need to evaluate the function's values throughout the entire interval, especially at the endpoints, to make the distinction.

Critical Points

Critical points of a function occur where the first derivative is either zero or undefined. These points are important because they are potential locations of local extremum (maximums or minimums) and inflection points.

To find a critical point, one would set the first derivative of the function equal to zero and solve for \( x \). However, not every critical point signifies a local extremum. For example, critical points can occur at the inflection points of a function where the concavity changes but the function does not achieve a local maximum or minimum.

Determining whether a critical point corresponds to a local maximum or minimum requires further analysis, often involving the second derivative test or the first derivative test where the sign of the slope changes are examined around the critical point.

Local Maximum

A local maximum of a function is a point where the function value is greater than all other function values in its immediate vicinity. At a local maximum point \( c \), the first derivative of the function \( f'(c) \) will be zero because the slope of the tangent to the curve at that point is horizontal.

When the function is concave down around that point, as indicated by a negative second derivative, that flat tangent represents the highest local value of the function, hence a local maximum. The existence of a local maximum does not necessarily imply the highest value in the given domain—this is where the concept of absolute maximum comes into play, comparing all function values within the domain.

Absolute Maximum

An absolute maximum is the highest function value over the entire domain. Unlike a local maximum, which is only concerned with values in a neighborhood around a point, the absolute maximum is the supreme value in the entire interval being considered.

In a closed interval \([a, b]\), we must evaluate the function at the critical points and at the interval's endpoints to determine the absolute maximum. For functions that are concave down throughout this interval, a local maximum found within the interval will be greater than or equal to the function values at the endpoints; hence, it will also be the absolute maximum.

In problems where we know the endpoints and the concavity between them, finding the function values at the endpoints and any critical points allows us to identify the absolute maximum with certainty.