Increasing and Decreasing Functions

Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing is an important concept in calculus and mathematics in general. It provides insight into the behavior of the function, particularly how the output changes as the input (or $x$-value) changes. These concepts are often used in optimization problems, curve sketching, and analyzing the growth or decay of quantities in various fields.

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1. Definition of Increasing and Decreasing Functions

• Increasing Function: A function $f(x)$ is said to be increasing on an interval if, for any two points $x_1$ and $x_2$ within that interval, where $x_1 < x_2$, it holds that:

  $$f(x_1) \leq f(x_2)$$

  This means that as $x$ increases, the function’s output (or $y$-value) either increases or remains the same.

• Strictly Increasing Function: If $f(x_1) < f(x_2)$ whenever $x_1 < x_2$, the function is strictly increasing. In this case, the output always increases as the input increases.

• Decreasing Function: A function $f(x)$ is said to be decreasing on an interval if, for any two points $x_1$ and $x_2$ within that interval, where $x_1 < x_2$, it holds that:

  $$f(x_1) \geq f(x_2)$$

  This means that as $x$ increases, the function’s output (or $y$-value) either decreases or remains the same.

• Strictly Decreasing Function: If $f(x_1) > f(x_2)$ whenever $x_1 < x_2$, the function is strictly decreasing. In this case, the output always decreases as the input increases.

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2. How to Determine Whether a Function is Increasing or Decreasing

To determine whether a function is increasing or decreasing, we typically use the first derivative of the function.

Step 1: Find the Derivative of the Function

The first derivative of a function $f(x)$, denoted $f'(x)$, gives information about the rate of change of the function. Specifically:

• If $f'(x) > 0$, the function is increasing at that point.
• If $f'(x) < 0$, the function is decreasing at that point.
• If $f'(x) = 0$, the function may have a local maximum, minimum, or inflection point.

Step 2: Analyze the Sign of the Derivative

• Increasing Function: If the derivative $f'(x) > 0$ on an interval, the function is increasing on that interval.
• Decreasing Function: If the derivative $f'(x) < 0$ on an interval, the function is decreasing on that interval.

Step 3: Find Critical Points and Test Intervals

Critical points occur where $f'(x) = 0$ or $f'(x)$ does not exist. After finding critical points, you can test the sign of the derivative in the intervals between the critical points. This method is called the First Derivative Test.

• To determine whether a function is increasing or decreasing on each interval, pick test points within each interval (the intervals determined by the critical points) and substitute them into $f'(x)$.
  • If $f'(x) > 0$ for a test point in the interval, the function is increasing on that interval.
  • If $f'(x) < 0$ for a test point in the interval, the function is decreasing on that interval.

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3. Example 1: Increasing and Decreasing Intervals of a Function

Consider the function $f(x) = x^3 - 3x^2 + 2x$.

Step 1: Find the derivative.

The first derivative of $f(x)$ is:

$$f'(x) = 3x^2 - 6x + 2$$

Step 2: Find critical points.

Set the derivative equal to 0 to find the critical points:

$$3x^2 - 6x + 2 = 0$$

Divide through by 3:

$$x^2 - 2x + \frac{2}{3} = 0$$

This equation does not factor nicely, so we solve it using the quadratic formula:

$$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(2/3)}}{2(1)} = \frac{2 \pm \sqrt{4 - 8/3}}{2} = \frac{2 \pm \sqrt{4 - 2.67}}{2}$$

This results in the critical points $x = 1.25$ and $x = -1.25$.

Step 3: Analyze the intervals.

Now, we have critical points at $x = 1.25$ and $x = -1.25$. We divide the real line into intervals based on these points: $(-\infty, -1.25)$, $(-1.25, 1.25)$, and $(1.25, \infty)$.

• For $x = -2$ (a point in the interval $(-\infty, -1.25)$): Substitute into the first derivative:

  $$f'(-2) = 3(-2)^2 - 6(-2) + 2 = 12 + 12 + 2 = 26$$

  Since $f'(-2) > 0$, the function is increasing on $(-\infty, -1.25)$.

• For $x = 0$ (a point in the interval $(-1.25, 1.25)$): Substitute into the first derivative:

  $$f'(0) = 3(0)^2 - 6(0) + 2 = 2$$

  Since $f'(0) > 0$, the function is increasing on $(-1.25, 1.25)$.

• For $x = 2$ (a point in the interval $(1.25, \infty)$): Substitute into the first derivative:

  $$f'(2) = 3(2)^2 - 6(2) + 2 = 12 - 12 + 2 = 2$$

  Since $f'(2) > 0$, the function is increasing on $(1.25, \infty)$.

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4. Example 2: Determining Decreasing and Increasing Intervals of a Function

Consider the function $f(x) = x^2 - 4x + 3$.

Step 1: Find the derivative.

The first derivative of $f(x)$ is:

$$f'(x) = 2x - 4$$

Step 2: Set the derivative equal to 0 and solve for $x$:

$$2x - 4 = 0$$ $$x = 2$$

Step 3: Test the intervals.

We have one critical point at $x = 2$. We divide the real line into two intervals: $(-\infty, 2)$ and $(2, \infty)$.

• For $x = 1$ (a point in the interval $(-\infty, 2)$): Substitute into the first derivative:

  $$f'(1) = 2(1) - 4 = -2$$

  Since $f'(1) < 0$, the function is decreasing on $(-\infty, 2)$.

• For $x = 3$ (a point in the interval $(2, \infty)$): Substitute into the first derivative:

  $$f'(3) = 2(3) - 4 = 2$$

  Since $f'(3) > 0$, the function is increasing on $(2, \infty)$.

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5. Summary of Key Concepts

Concept	Description
Increasing Function	$f(x)$ is increasing if $f'(x) > 0$ on an interval.
Decreasing Function	$f(x)$ is decreasing if $f'(x) < 0$ on an interval.
Critical Points	Points where $f'(x) = 0$ or $f'(x)$ does not exist, potential points for local maxima, minima, or inflection.
First Derivative Test	A method to determine whether a function is increasing or decreasing on an interval.
Strictly Increasing/Decreasing	If the derivative is strictly positive (or negative), the function is strictly increasing (or decreasing).

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Conclusion

• To determine whether a function is increasing or decreasing, find its first derivative $f'(x)$, and analyze its sign over intervals.
• Increasing functions have $f'(x) > 0$ over the interval, while decreasing functions have $f'(x) < 0$ over the interval.
• Use critical points and test intervals to understand the behavior of the function.