Increasing and Decreasing Functions

Increasing Functions: A function $f$ is said to be increasing on an interval if, for any two numbers $x_1$ and $x_2$ in the interval, $x_1 < x_2$ implies that $f(x_1) \le f(x_2)$. This means as you move from left to right along the x-axis, the graph of the function moves upward or stays level.

Decreasing Functions: Conversely, a function $f$ is decreasing on an interval if $x_1 < x_2$ implies $f(x_1) \ge f(x_2)$. In this case, the graph moves downward or stays level as the input values increase.

Strict Monotonicity: A function is strictly increasing if $f(x_1) < f(x_2)$ and strictly decreasing if $f(x_1) > f(x_2)$. This excludes constant horizontal segments, ensuring the function is always moving in one direction.

Monotonic Functions: A function that is either entirely non-increasing or entirely non-decreasing over its entire domain is referred to as a monotonic function.

The First Derivative as Slope: The derivative $f'(x)$ represents the instantaneous rate of change or the slope of the tangent line at any point $x$. If the slope is positive, the function must be rising; if negative, it must be falling.

Test for Increasing Functions: If $f'(x) > 0$ for every $x$ in an open interval $(a, b)$, then the function $f$ is strictly increasing on the closed interval $[a, b]$, provided $f$ is continuous on that interval.

Test for Decreasing Functions: If $f'(x) < 0$ for every $x$ in an open interval $(a, b)$, then the function $f$ is strictly decreasing on the closed interval $[a, b]$.

Constant Functions: If $f'(x) = 0$ for all $x$ in an interval, the function is constant, meaning its graph is a horizontal line with no change in vertical position.

Step 1: Find the Derivative: Calculate the first derivative $f'(x)$ of the given function using standard differentiation rules such as the power, product, or chain rules.

Step 2: Identify Critical Points: Determine the values of $x$ where $f'(x) = 0$ or where $f'(x)$ is undefined. These points, along with any points where the function itself is undefined, serve as the boundaries for potential changes in direction.

Step 3: Create Test Intervals: Use the critical points to divide the domain of the function into several open intervals. For example, if critical points are $c_1$ and $c_2$, the intervals would be $(-\infty, c_1)$, $(c_1, c_2)$, and $(c_2, \infty)$.

Step 4: Evaluate the Sign: Choose a 'test value' from within each interval and substitute it into $f'(x)$. The sign of the result (positive or negative) determines the behavior of the function for the entire interval.

Step 5: Conclude Behavior: Summarize the intervals where the function is increasing ($f'(x) > 0$) and where it is decreasing ($f'(x) < 0$).

Critical Points vs. Partition Points: While critical points are where the derivative is zero or undefined, partition points also include values where the original function is undefined (like vertical asymptotes). Both must be used to separate intervals for testing.

Strict vs. Non-Strict: In many calculus contexts, 'increasing' is used to mean 'strictly increasing'. However, if a function has a flat section where $f'(x)=0$ for a range of $x$, it is non-strictly increasing.

Assuming $f'(x)=0$ implies a turn: Just because the derivative is zero at a point doesn't mean the function changes direction. For example, $f(x) = x^3$ has $f'(0) = 0$, but the function is increasing both before and after $x=0$.

Misinterpreting the Sign of $f(x)$: A common mistake is looking at whether the function values $f(x)$ are positive or negative. The direction of the function depends entirely on the sign of the derivative $f'(x)$, not the function itself.

Ignoring Discontinuities: If a function has a vertical asymptote, the behavior can change across that asymptote even if there is no critical point there. Always include domain restrictions in your interval analysis.