Problem 1 - Calculus

Question

In Problems \(I-10\), use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ f(x)=3 x+3 $$

Step-by-Step Solution

Answer

The function \(f(x) = 3x + 3\) is increasing on \((- fty, fty)\).

Step 1: Differentiate the function

To find where the function is increasing or decreasing, we first need to find the derivative of the function. The function given is \(f(x) = 3x + 3\). The derivative of \(f(x)\) with respect to \(x\) is:\[f'(x) = 3\]

Step 2: Analyze the derivative

According to the Monotonicity Theorem, a function is increasing where its derivative is positive and decreasing where its derivative is negative. Here, \(f'(x) = 3\), which is a constant positive value.

Step 3: Determine intervals of increase and decrease

Since \(f'(x) = 3\) is positive for all \(x\) in \((-fty, fty)\), the function \(f(x) = 3x + 3\) is increasing on \((-fty, fty)\). There are no intervals where \(f(x)\) is decreasing because the derivative is never negative.

Key Concepts

Derivative

The concept of a derivative is fundamental in understanding the behavior of functions. When we talk about the derivative of a function, we're looking for a mathematical representation that tells us how the function changes at any given point. In simpler terms, it reveals the rate at which the function's output value y changes as the input value x changes. For the function in question, \(f(x) = 3x + 3\), finding the derivative involves applying basic differentiation rules. The derivative is computed as \(f'(x) = 3\), which is a constant derivative. This uniform value indicates that the slope of the tangent line to the graph of the function is always the same regardless of x. It expresses a consistent rate of change across the entire domain of the function. In this particular example, because \(f'(x)\) is a constant positive number, it tells us that the function increases uniformly as x increases. The derivative here serves as a tool to explore further properties of the function, such as where it is increasing or decreasing.

Increasing Functions

Increasing functions are an important concept in calculus and analysis that describe where a function's output rises as the input increases. For a function to be defined as increasing on an interval, its derivative must be positive for all points within that interval. In the case of the function \(f(x) = 3x + 3\), we find that the derivative \(f'(x) = 3\) is positive for all values of x. Hence, this means that the function does not only increase at some points but is increasing across its entire domain: \((-\infty, \infty)\).

• This is due to the derivative being a constant positive value.
• Such functions are often referred to as having a linear growth because the rate of increase does not change - it's constant.
• Whenever x increases by a unit, y increases by 3 units consistently.

Thus, through the derivative's analysis, we can conclude without ambiguity that \(f(x)\) is strictly increasing wherever it is defined.

Decreasing Functions

Unlike increasing functions, decreasing functions have a negative rate of change. To identify a decreasing function on an interval, the derivative of the function has to be negative for every point in that interval. However, when analyzing our function \(f(x) = 3x + 3\), the derivative \(f'(x) = 3\) never becomes negative.

• This means there are no segments on its domain where the function decreases.
• Instead, because the derivative is always positive, it confirms that the function is perpetually increasing across all x values.
• Consequently, \(f(x)\) spans its entire domain without any intervals of decline.

Therefore, for this problem, considering decreasing functions is straightforward since there simply aren't any. Understanding this helps students solidify their grasp of how derivatives inform us about the behavior of different function types.