Problem 1 - Calculus Early Transcendentals

Question

In each part, sketch the graph of a function \(f\) with the stated properties, and discuss the signs of \(f^{\prime}\) and \(f^{\prime \prime} .\) (a) The function \(f\) is concave up and increasing on the interval \((-\infty,+\infty)\) (b) The function \(f\) is concave down and increasing on the interval \((-\infty,+\infty) .\) (c) The function \(f\) is concave up and decreasing on the interval \((-\infty,+\infty) .\) (d) The function \(f\) is concave down and decreasing on the interval \((-\infty,+\infty) .\)

Step-by-Step Solution

Answer

(a) Increasing, concave up. (b) Increasing, concave down. (c) Decreasing, concave up. (d) Decreasing, concave down.

Step 1: Understanding Concave Up and Increasing Function

For part (a), we want a function that is concave up and increasing. Concavity refers to the direction of the curve's bend, and an increasing function means it rises as it moves from left to right. A standard function with these properties is a parabola opening upwards, such as \( f(x) = x^2 \). Here, \( f'(x) = 2x > 0 \) (increasing) and \( f''(x) = 2 > 0 \) (concave up) for \( x \in (-\infty, +\infty) \).

Step 2: Understanding Concave Down and Increasing Function

For part (b), the function should be concave down and increasing. A function like \( f(x) = 1 - e^{-x} \) fits, as it bends downwards but still increases. Calculation of derivatives gives \( f'(x) = e^{-x} > 0 \) (increasing) and \( f''(x) = -e^{-x} < 0 \) (concave down) for \( x \in (-\infty, +\infty) \).

Step 3: Understanding Concave Up and Decreasing Function

For part (c), the function is concave up and decreasing. An example is \( f(x) = -x^2 \), which opens upwards but decreases as \( x \) moves left to right. Here, \( f'(x) = -2x < 0 \) (decreasing) if \( x < 0 \) and \( f''(x) = -2 < 0 \) must be positive, so a function like \( f(x) = -\frac{1}{x} \) where \( f'(x) = \frac{1}{x^2} > 0 \) could fit after revising signs for decreasing structure.

Step 4: Understanding Concave Down and Decreasing Function

Finally, for part (d), choose a function that is concave down and decreases. The function \( f(x) = -x - e^{-x} \) suits this, as it declines while bending downwards. Here, \( f'(x) = -1 + e^{-x} < 0 \) (decreasing) and \( f''(x) = -e^{-x} < 0 \) (concave down) for \( x \in (-\infty, +\infty) \).

Key Concepts

Concavity in Calculus

Concavity in calculus helps us understand a function’s behavior by describing how it curves. A function is considered "concave up" if the curve bends upwards, similar to a smile, while it is "concave down" if it bends downwards, like a frown. The concavity of a function is determined by the second derivative.

• If the second derivative, \( f''(x) \), is positive over an interval, the function is concave up there.
• Conversely, if \( f''(x) \) is negative, the function is concave down.

Knowing whether a function is concave up or down can help predict how the curve behaves between critical points and whether they are minima or maxima.

Derivative Signs

The sign of a derivative provides essential insight into a function's nature. The first derivative, \( f'(x) \), indicates how a function is changing at any given point.

• When \( f'(x) \) is positive, the function increases as \( x \) increases.
• If \( f'(x) \) is negative, the function decreases.
• A zero value of \( f'(x) \) suggests potential extrema (maximum or minimum points).

By evaluating the sign of the first derivative, we can draw vital conclusions about the increasing or decreasing nature of the function across its domain.

Increasing and Decreasing Functions

Functions may seem simple enough: they're either increasing, meaning they go up as \( x \) moves from left to right, or decreasing, meaning they go down.

• For increasing functions, each subsequent \( f(x) \) is greater than the previous.
• In decreasing functions, each \( f(x) \) is less than the previous when \( x \) advances.

The relationship between derivatives and the gradient of a function is crucial. Positive \( f'(x) \) ensures that the function is rising, while a negative \( f'(x) \) ensures a falling function. This understanding aids in sketching graphs and solving problems related to function behaviors effectively.