Question

A function \(f\) has derivative \(f^{\prime}(x)=(\sin x+2) e^{x^{2}+1},\) where \(f^{\prime}(x)>1\) for all \(x .\) Is \(f\) increasing, decreasing, or can we not tell from the given information?

Short answer

The function \( f \) is increasing for all \( x \).

Step-by-step solution

Understanding the Given Information

We are given that the derivative of the function \( f \) is \( f^{\prime}(x)=(\sin x+2) e^{x^{2}+1} \). Additionally, we know that \( f^{\prime}(x) > 1 \) for all \( x \). Our goal is to determine the behavior of the function \( f \) based on its derivative.

Analyzing the Expression for the Derivative

The derivative \( f^{\prime}(x) = (\sin x+2) e^{x^{2}+1} \) consists of two parts: \( \sin x + 2 \) and \( e^{x^{2}+1} \). The term \( e^{x^{2}+1} \) is always positive for all \( x \), as the exponential function is always positive.

Evaluating \( \sin x + 2 \)

\( \sin x \) has a range of \([-1, 1]\). Therefore, the range for \( \sin x + 2 \) is \([1, 3]\). Since this term is always greater than or equal to \(1\), the expression \( (\sin x + 2) e^{x^{2}+1} \) is always positive.

Concluding the Behavior of \( f \)

Since \( f^{\prime}(x) > 1 \) for all \( x \), the derivative is always positive. A positive derivative implies that the function \( f \) is increasing for all \( x \).

Key concepts

Understanding Derivatives

In calculus, a derivative represents how a function changes as its input changes. It's like the speedometer in a car; it tells you how fast the function's value is changing at any given point. Derivatives can provide a wide array of information about a function's behavior, such as where it is increasing or decreasing.

Key aspects of derivatives include:

• The notation: The derivative of a function \( f(x) \) is often written as \( f'(x) \) or \( \frac{df}{dx} \), indicating the rate of change of the function \( f \) with respect to \( x \).
• Interpreting derivatives: A positive derivative generally indicates that a function is increasing, while a negative derivative indicates a decreasing function.
• Calculating derivatives: There are rules, like the product rule and chain rule, that help us find the derivative for more complex functions.

In this exercise, we use the derivative \( f'(x)=(\sin x+2) e^{x^{2}+1} \) to determine the behavior of the function \( f(x) \). With \( f'(x) > 1 \) for all \( x \), we're assured that the function is increasing everywhere.

Exploring Increasing Functions

An increasing function is one that rises as you move along the x-axis. In mathematical terms, this means that when \( x_1 < x_2 \), then \( f(x_1) < f(x_2) \). If the derivative of a function is positive at all points, this tells us the function is increasing everywhere along its domain.

Features of increasing functions include:

• Positive derivative: If \( f'(x) > 0 \) for all \( x \), the function is strictly increasing.
• Gradual or steep ascent: Depending on how large \( f'(x) \) is, the slope or steepness of the increase will vary.
• Visually recognizable: On a graph, an increasing function will consistently move upwards from left to right.

In our context, knowing that \( f'(x) = (\sin x + 2) e^{x^2+1} > 1 \) for all \( x \) confirms that the function \( f \) is increasing for all values of \( x \). This is due to the positive nature of this derivative expression across all points.

Characteristics of Exponential Functions

Exponential functions are powerful mathematical tools, widely used to model growth and decay processes in real-world scenarios. The general form of an exponential function is \( a^x \), where \( a \) is a positive constant.

Here are some properties of exponential functions:

• Rapid growth: Exponential functions grow much faster than linear or polynomial functions as \( x \) increases.
• Always positive: In its standard form, an exponential function is positive for all values of \( x \).
• Differentiability: Exponential functions are smooth and continuous, making them easy to differentiate and integrate.
• Graph trends: The graph of an exponential function typically starts slowly and then rapidly increases if the base is greater than 1.

In the given derivative \( f'(x) = (\sin x + 2) e^{x^{2}+1} \), \( e^{x^{2}+1} \) is an exponential term. This not only ensures that it’s always positive, but also contributes to the increasing nature of the function \( f \) due to its multiplying effect on the entirety of the expression. This property is essential for understanding why the derivative remains greater than 1, ensuring that the function is always increasing.