Problem 10 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=5-3 x $$

Step-by-Step Solution

Answer

There are no critical numbers for this function. The function \(f(x)=5-3x\) is decreasing on the interval \(-\infty < x < \infty\).

Step 1: Find the Derivative

The derivative of the function needs to be found. The derivative of a function tells us the rate of change of the function at any point. In this case, the derivative of \(f(x)=5-3x\) is \(f'(x)=-3\).

Step 2: Find the Critical Numbers

The next step is identifying the critical numbers, which are the points at which the derivative of the function is zero or undefined. The derivative \(f'(x)=-3\) is a constant and never undefined. So there are no critical numbers for this function.

Step 3: Determine the intervals of Increase and Decrease

Since the derivative of the function is negative (-3) everywhere for all real number x, the function \(f(x)=5-3x\) is decreasing over the interval \(-\infty < x < \infty\).

Key Concepts

Derivative of a Function

A derivative of a function provides us with the rate at which the function's value changes at any given point. This is like knowing the speed of a car at any given moment, it tells us how fast or slow the function's output changes as we tweak the input. For the function provided, which is a linear function, the derivative is particularly simple. When we differentiate the function \(f(x) = 5 - 3x\), we end up with \(f'(x) = -3\). This constant derivative means that for every unit increase in \(x\), \(f(x)\) decreases by 3. This negative sign in the derivative denotes a consistent decrease in the function’s value as \(x\) increases, leading us to the next concept of increase and decrease intervals.

Intervals of Increase and Decrease

Once we have the derivative, determining where the function increases or decreases becomes straightforward. Typically, one finds where the derivative is zero or changes signs, which are our critical numbers. For our given function, the derivative \(f'(x) = -3\) is always negative, meaning the function is always decreasing.

• There are no critical numbers because the derivative is constant and never changes sign.
• As a result, the function decreases across the entire real number line, from \(-\infty < x < \infty\).

This is because a negative derivative indicates a downward slope, with a consistent rate of decrease no matter what the value of \(x\) is.

Graphing Functions

Graphing helps us visualize what we have calculated. It is like seeing the terrain on a map after calculating the journey steepness using mathematical concepts. A linear function such as \(f(x) = 5 - 3x\) is represented by a straight line.

• The graph of this function will have a constant downward slope because its derivative \(f'(x) = -3\) is constant and negative.
• The y-intercept of the graph is \(5\) since when \(x = 0\), \(f(x) = 5\).
• The downward slope simply means that for every step to the right, the graph steps 3 units down, keeping with the rule of decrease we discovered.

By graphing the function using a utility tool, we can see this linear graph making a consistent downward path from the top left to bottom right, aligning perfectly with the function's decrease across all \(x\) values.