Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=3 x+5 $$

Step-by-Step Solution

Answer

The function \(f(x) = 3x + 5\) is increasing on the interval \((-\infty, \infty)\) and there are no decreasing intervals.

Step 1: Find the derivative of the function

We have the function \(f(x) = 3x + 5\). Find the derivative of this function. We'll denote the derivative as \(f'(x)\). Using the power rule, we get: \[ f'(x) = 3 \]

Step 2: Analyze the sign of the derivative

In this case, the derivative, \(f'(x) = 3\), is a constant value and is positive. Since the derivative is always positive, the function is increasing on its entire domain.

Step 3: State the interval for increasing and decreasing

As the derivative is positive for all values of x, the function is increasing on the interval \((-\infty, \infty)\), and there are no decreasing intervals. So, the function \(f(x) = 3x + 5\) is increasing on the interval \((-\infty, \infty)\) and there are no decreasing intervals.

Key Concepts

Derivative

A derivative is a fundamental concept in calculus that measures how a function's output changes as its input changes. Think of it as the rate of change or slope of the function at any given point.

For example, if the function is a graph of a line, the derivative will give you the steepness of this line. A high derivative means a steep slope, while a low derivative indicates a gentle slope.

To find the derivative, you often use rules such as the power rule, product rule, or chain rule, depending on the function. In our example, the function is linear, so the derivative is constant, meaning the rate of change does not vary across different values of x.

Increasing Function

An increasing function means that as you move along the graph from left to right, the value of the function rises.

If you imagine walking up a hill, an increasing function would be like steadily moving upwards. This happens when the derivative of the function is positive.

• A positive derivative indicates that the slope of the function is rising.
• If the derivative is positive on an interval, the function is increasing on that interval.

In our specific problem, the derivative we calculated (\( f'(x) = 3 \)) is positive for all x. Thus, the function increases over its entire domain, \((-\infty, \infty)\).

Decreasing Function

Conversely, a decreasing function is one where, as you go from left to right, the function’s value falls. Imagine it like going downhill.

A function is considered decreasing on an interval where its derivative is negative.

• A negative derivative means the slope is downward, leading the function to decrease.
• If the derivative is negative over an interval, the function is decreasing over that interval.

In the problem at hand, since the derivative \( f'(x) = 3 \) is never negative, there are no intervals on which the function is decreasing. This tells us the whole story of our function being increasing consistently.

Power Rule

The power rule is a straightforward and essential tool for finding derivatives, particularly useful when dealing with polynomials.

It tells you how to differentiate expressions of the form \( x^n \). To use the power rule:

• Bring down the exponent as a coefficient.
• Subtract one from the original exponent.

For example, if you have \( x^3 \), the derivative using the power rule is \( 3x^2 \).

In our exercise, applying the power rule to a linear function, \( f(x) = 3x + 5 \), results in \( f'(x) = 3 \), as the derivative of a constant term, \( 5 \), is \( 0 \), and \( x \) becomes \( 1 \) after differentiation.