Question

Is it possible? Determine whether the following properties can be satisfied by a function that is continuous on \((-\infty, \infty)\). If such a function is possible, provide an example or a sketch of the function. If such a function is not possible, explain why. a. A function \(f\) is concave down and positive everywhere. b. A function \(f\) is increasing and concave down everywhere. c. A function \(f\) has exactly two local extrema and three inflection points. d. A function \(f\) has exactly four zeros and two local extrema.

Short answer

If so, provide an example or a sketch of the function. If not, explain why it is not possible. a. Be concave down and positive everywhere. b. Be increasing and concave down everywhere. c. Have two local extrema and three inflection points. d. Have four zeros and two local extrema. Answer: a. Yes, it is possible. An example of such a function is \(f(x) = e^{-x^2}\). b. No, it is not possible for a function to be both increasing and concave down everywhere. c. Yes, it is possible. An example of such a function is \(f(x) = x^4 - 4x^2 + 4\), with local extrema at \(x= -1, 1\) and inflection points at \(x= -\sqrt{2},0,\sqrt{2}\). d. Yes, it is possible. An example of such a function is \(f(x) = x^4 - 4x^3 - 2x^2 + 16x\), with zeros at \(x = 0, 2, -2, 4\) and local extrema at \(x = \frac{4}{3}, 2\).

Step-by-step solution

a. Concave down and positive everywhere

Yes, it is possible. A continuous function can be concave down and positive everywhere. One example of such a function is \(f(x) = e^{-x^2}\). This function is always positive because the exponent is always negative, and it is concave down everywhere because its second derivative, \(f''(x) = e^{-x^2}(4x^4 - 4x^2 + 2)\), is negative for all \(x\).

b. Increasing and concave down everywhere

No, it is not possible for a function to be both increasing and concave down everywhere. A function that is concave down everywhere has a negative second derivative, which means its slope would need to decrease as \(x\) increases. However, an increasing function has a positive first derivative and thus its slope would need to increase as \(x\) increases. These two conditions cannot be satisfied simultaneously.

c. Two local extrema and three inflection points

Yes, it is possible. For a function to have exactly two local extrema and three inflection points, we require a function with at least a fourth-degree polynomial. An example of such a function is \(f(x) = x^4 - 4x^2 + 4\). - This function has local extrema at \(x= -1, 1\), and inflection points at \(x= -\sqrt{2},0,\sqrt{2}\). - To see this, we compute the first and second derivative of \(f(x)\): - First derivative: \(f'(x) = 4x^3 - 8x\) - Second derivative: \(f''(x) = 12x^2 - 8\) - The local extrema occur when \(f'(x) = 0\), and inflection points occur when \(f''(x) = 0\).

d. Four zeros and two local extrema

Yes, it is possible. For a function to have exactly four zeros and two local extrema, we can use a fourth-degree polynomial function with a graph that intersects the x-axis four times and has two local extrema between them. An example of such a function is \(f(x) = x^4 - 4x^3 - 2x^2 + 16x\). - This function has zeros at \(x = 0, 2, -2, 4\) and local extrema at \(x = \frac{4}{3}, 2\). - To see this, we compute the first and second derivative of \(f(x)\): - First derivative: \(f'(x) = 4x^3 - 12x^2 - 4x + 16\) - Second derivative: \(f''(x) = 12x^2 - 24x - 4\) - The zeros occur when \(f(x) = 0\), and the local extrema occur when \(f'(x) = 0\).

Key concepts

Concave Down Function

Understanding a 'concave down function' is essential for analyzing the shape and behavior of graphs. When we say a function is concave down, we mean that as you move along the graph from left to right, the line or curve is curving downwards, like the inside of a bowl facing up or an upside down arch.

A key factor that determines the concavity of a function is the sign of its second derivative. If the second derivative of a function is negative over a certain interval, then the function is concave down on that interval. For example, the function given in the exercise, \(f(x) = e^{-x^2}\), is concave down because its second derivative is negative for all \(x\). This implies that the gradient of the tangent to the function's graph decreases as \(x\) increases.

Visualizing a concave down graph, one should imagine a downhill slope that steepens continuously or a surface that curves downward, with the gradient becoming more negative as you progress. It is the shape of the graph that allows us to categorize the function as concave down, a fundamental concept in calculus with applications in economics, physics, and engineering.

Increasing Function

A function is called an 'increasing function' if, for any two points on the graph where \(x_1 < x_2\), we have \(f(x_1) \leq f(x_2)\). This simply means that as \(x\) increases, so does \(f(x)\). An increasing function has a slope that never becomes negative—that is, its first derivative is always positive or zero.

The exercise highlights the impossibility for a function to be both increasing and concave down everywhere. An increasing function naturally has a positive first derivative, suggesting that its graph is rising. However, a concave down graph has a reducing slope as you read from left to right. This contradiction makes it impossible for a function to display these two properties simultaneously across its entire domain.

Understanding increasing functions is pivotal in various fields that study growth, trends, and positive changes, like population dynamics, stock market analysis, and many more. It's instrumental in establishing whether a given variable or quantity is experiencing a consistent rise over time.

Local Extrema

When we speak of 'local extrema' of a function, we are referring to the points on the graph where the function reaches a local minimum or local maximum value. These points represent a peak or trough in a small interval around the point, meaning that there’s no higher (for maxima) or lower (for minima) value close to that point.

To find local extrema, we look for points where the first derivative of the function equals zero and changes signs—a condition met by a function given in the exercise, \(f(x) = x^4 - 4x^2 + 4\). The change in sign of the first derivative from positive to negative indicates a local maximum, while the opposite change, from negative to positive, indicates a local minimum.

Local extrema are crucial in optimization problems where the objective is to find the maximum or minimum value of a function within a certain range. This concept is widely applied in business for maximizing profits or minimizing costs, in engineering for optimizing designs, and in everyday life decisions like finding the most efficient routes.

Inflection Points

Inflection points mark the spots on the graph of a function where the curvature changes direction. In simpler terms, an inflection point is where the graph transitions from being concave upward to concave downward, or vice versa. This change in concavity is detected when the second derivative of the function changes signs.

For example, in the function \(f(x) = x^4 - 4x^2 + 4\) provided in the exercise, the inflection points occur where the second derivative, \(f''(x) = 12x^2 - 8\), is equal to zero. To confirm that these points are inflection points, we investigate the sign change of the second derivative around those points.

The existence of inflection points is highly informative in graph analysis, as they provide critical information about the function's shape and the dynamics of its behavior. They are used in various applications that require an understanding of how the rate of change is altering – for instance, in economics to find points of diminishing returns or in physics to understand the motion under variable forces.