Monotonicity Based Problems - Interval of Increase and Decrease

Monotonicity in mathematics refers to the behavior of a function as either consistently increasing or decreasing. Understanding the intervals of increase and decrease of a function is crucial for solving a variety of problems in calculus and its applications.

Understanding Monotonic Functions

A function is said to be monotonic if it is either entirely non-increasing or non-decreasing throughout its domain. More specifically:

• A function $f(x)$ is increasing on an interval if for any two numbers $a$ and $b$ within the interval, $a < b$ implies $f(a) < f(b)$.
• A function $f(x)$ is decreasing on an interval if for any two numbers $a$ and $b$ within the interval, $a < b$ implies $f(a) > f(b)$.

Determining Monotonicity

To determine where a function is increasing or decreasing, we use its first derivative, $f'(x)$. The sign of the derivative provides information about the function's slope and, consequently, its monotonicity.

Condition	Derivative Sign	Monotonicity
Increasing	$f'(x) > 0$	Function is going up
Decreasing	$f'(x) < 0$	Function is going down
Constant	$f'(x) = 0$	Function is neither increasing nor decreasing

Steps to Find Intervals of Increase and Decrease

1. Find the derivative: Compute the first derivative $f'(x)$ of the function $f(x)$.
2. Determine critical points: Solve $f'(x) = 0$ to find critical points where the function's slope may change.
3. Test intervals: Choose test points in the intervals determined by the critical points and evaluate $f'(x)$ at these points to determine the sign of the derivative.
4. Draw conclusions: Based on the sign of $f'(x)$, determine where the function is increasing or decreasing.

Example Problem

Let's consider the function $f(x) = x^3 - 3x^2 + 4$.

Step 1: Find the derivative

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

Step 2: Determine critical points

Solve $f'(x) = 0$:

$$ 3x^2 - 6x = 0 $$ $$ x(3x - 6) = 0 $$ $$ x = 0 \text{ or } x = 2 $$

Step 3: Test intervals

We have two critical points, $x = 0$ and $x = 2$, which divide the real line into three intervals: $(-\infty, 0)$, $(0, 2)$, and $(2, \infty)$. We choose test points from each interval, say $x = -1$, $x = 1$, and $x = 3$, and evaluate $f'(x)$:

• For $x = -1$: $f'(-1) = 3(-1)^2 - 6(-1) = 3 + 6 = 9 > 0$
• For $x = 1$: $f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3 < 0$
• For $x = 3$: $f'(3) = 3(3)^2 - 6(3) = 27 - 18 = 9 > 0$

Step 4: Draw conclusions

Based on the sign of $f'(x)$ in the test points, we can conclude:

• The function is increasing on the interval $(-\infty, 0)$ and $(2, \infty)$.
• The function is decreasing on the interval $(0, 2)$.

Graphical Representation

A graph of the function can also help visualize the intervals of increase and decrease. Here, the slope of the tangent line to the graph corresponds to the sign of the derivative.

Important Points to Remember

• A function can be increasing or decreasing in different intervals.
• The derivative test gives us a method to find these intervals.
• Critical points are where the derivative is zero or undefined.
• Always test intervals with points that are easy to compute.

Conclusion

Understanding monotonicity and the intervals of increase and decrease is essential for analyzing functions and their behavior. By following the steps outlined above and practicing with various functions, one can master the concept of monotonicity and apply it to solve complex problems in calculus.