Problem 114 - College Algebra Essentials

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.

Step-by-Step Solution

Answer

The statement does not make sense because a point \(f(a)\) on function \(f(x)\) is a relative maximum if the function is increasing on the interval to the left of \(a\) and decreasing on the interval to the right of \(a\). In the given statement, the function \(f(x)\) is decreasing on the interval to the left of \(a\) and increasing on the interval to the right of \(a\), which is the characteristic of a relative minimum, not a relative maximum.

Step 1: Understanding the problem

The given statement argues that if a function \(f(a)\) is decreasing before \(a\) (on \((-\infty, a)\)) and increasing after \(a\) (on \((a, \infty)\)), then \(f(a)\) must be a relative maximum.

Step 2: Analyzing the statement

A relative maximum as mentioned earlier, is a point that is higher than the points immediately to the left and right of it. For a point to be a relative maximum, the function should be increasing on the interval to the left of the point (i.e., \(f(x)\) should be increasing on \((-\infty, a)\)) and decreasing on the interval to the right of the point (i.e., \(f(x)\) should be decreasing on \((a, \infty)\)).

Step 3: Verifying the statement

In the given statement, the function \(f(x)\) is said to be decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\), which is the description for a relative minimum point, not a relative maximum point. Hence, the statement does not make sense.

Key Concepts

Decreasing Function

A decreasing function is one where, as you move along the x-axis, the y-values reduce. This can be visualized as moving downhill on a graph. As we progress from left to right, each output value is smaller than the previous one. This indicates that the function is decreasing.
To elaborate:

• Imagine a slope that continuously declines as you proceed.
• Mathematically, this means that for any two points, if the first point, say \(x_1\), is less than the second point \(x_2\); then \(f(x_1) > f(x_2)\).

Understanding where a function decreases helps identify crucial points in its graph, like possible minima.

Increasing Function

When we talk about an increasing function, we are referring to one where, as you move along the x-axis to the right, the y-values also go up. Picture this like hiking up a hill. With every step forward, you are gaining altitude.
Here's how this idea works:

• For any two values on the x-axis, if the first one, \(x_1\), is less than the second one, \(x_2\), then \(f(x_1) < f(x_2)\).
• Essentially, as you move from left to right across the graph, you are moving upwards.

Recognizing where a function increases is fundamental in finding patterns and critical points such as maxima.

Relative Minimum

A relative minimum is a special point on a graph of a function where the y-value is lower than its immediate neighboring y-values on both sides. It's like finding a local valley. If you're walking through a landscape, this would be the lowest point between two upward slopes.
Understanding relative minimum:

• The function is decreasing before this point, reaching a low, and then it starts increasing after passing the minimum point.
• To identify a relative minimum, check that the slope to its left drops, and then rises on its right.
• Formally, \(f(x)\) is decreasing on \((a - \varepsilon, a)\) and increasing on \((a, a + \varepsilon)\) for some small positive number \(\varepsilon\).

Recognizing a relative minimum helps in understanding and predicting the behavior of a graph.