Problem 1 - Calculus A New Horizon

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

Use quadratic, linear, and cubic functions for different properties.

Step 1: Analyze Part (a)

For part (a), we need a continuous function that is always concave up and has one relative extremum (either a maximum or minimum). A suitable function is a parabola opening upwards, such as \( f(x) = x^2 \). This function is concave up everywhere and has a minimum at \(x = 0\).

Step 2: Analyze Part (b)

For part (b), we seek a continuous function concave up everywhere but with no relative extrema. A linear function fits this description since it is concave up and does not change direction. A function such as \( f(x) = 2x + 3 \) is adequate as it is straight and doesn't have any maxima or minima.

Step 3: Analyze Part (c)

For part (c), we need a continuous function with two relative extrema and that tends to positive infinity as \(x\) goes to positive infinity. A cubic function can fit these criteria, like \( f(x) = x^3 - 3x \), which has a local maximum and minimum and becomes large as \(x\) increases.

Step 4: Analyze Part (d)

For part (d), we need a continuous function with two relative extrema, which tends to negative infinity as \(x\) goes to positive infinity. We can consider a negative cubic function, such as \( f(x) = -x^3 + 3x \). This function also has one local maximum and one local minimum, and it decreases as \(x\) goes to positive infinity.

Key Concepts

Concavity

In mathematics, concavity is an important concept that helps us understand the bending behavior of a graph. A function is considered to be concave up if its second derivative is greater than zero over the interval being considered. This means that the graph of the function is shaped like a "cup," always curving upwards. Consider the function \(f(x) = x^2\).

This function is concave up everywhere since its second derivative \(f''(x) = 2\) is positive. Concavity helps us predict the general shape of the graph even if we don't know every detail of the function. If a function is concave up on a given interval, any tangent line drawn to the graph will lie below the curve.

Understanding concavity is essential for interpreting graphs because it influences how we determine points of extrema, such as local maxima and minima, which are critical for analyzing and sketching functions precisely.

Relative Extrema

Relative extrema refer to the local peaks (maxima) or valleys (minima) on a graph. They are points where a function changes direction, that is, from increasing to decreasing, or vice versa. Finding these points involves using the first derivative of the function. Consider a function like \(f(x) = x^3 - 3x\).

To find the relative extrema, we set the first derivative \(f'(x) = 3x^2 - 3\) to zero, solving for \(x\). This gives us potential points for relative extrema. After solving, we can test each root with the second derivative test to confirm whether it is a maximum or minimum.

These critical points are where the slope of the tangent is zero, meaning the graph flattens out momentarily. Additionally, the behavior of the graph at these points is influenced by concavity, which helps us further classify the extrema as maxima or minima. Mastering these skills is fundamental for drawing accurate function curves.

Behavior at Infinity

When analyzing functions, understanding how they behave as they approach infinity is crucial. This tells us about the "tail ends" of the graph. Essentially, it's about identifying the direction the graph heads towards as \(x\) becomes very large, either positively or negatively. For instance, the function \(f(x) = x^3 - 3x\) behaves differently than \(f(x) = -x^3 + 3x\) at infinity.

We specifically look at the highest degree term in the polynomial to determine behavior at infinity. If the leading coefficient is positive, the function tends towards positive infinity. Conversely, if the leading coefficient is negative, it tends towards negative infinity.

Understanding this concept is particularly important when graphing, as it enables us to draw the curve with the correct end-behavior. This insight helps us anticipate the function's directions without graphing it fully, which is essential for efficiently sketching and interpreting complex functions.