Applications of Derivatives: Monotonicity (Increasing/Decreasing Functions)

Increasing and Decreasing Functions: Definitions

The concept of a function being increasing or decreasing, collectively known as monotonicity, describes the overall trend of the function's output values as the input variable increases. This is a fundamental property used in analyzing the behavior and shape of a function's graph.

Definitions on an Interval $I$

Let $f$ be a real-valued function defined on an interval $I$ of real numbers. The interval $I$ can be any type of interval: open $(a, b)$, closed $[a, b]$, half-open $[a, b)$ or $(a, b]$, or infinite $(-\infty, b)$, $(a, \infty)$, $(-\infty, \infty)$.

The definitions of increasing and decreasing functions are based on comparing function values for different inputs within the interval.

1. Strictly Increasing Function: A function $f$ is said to be strictly increasing on the interval $I$ if for any two distinct numbers $x_1$ and $x_2$ chosen from $I$, whenever $x_1$ is less than $x_2$, the corresponding function value $f(x_1)$ is strictly less than $f(x_2)$.

   Formally: $f$ is strictly increasing on $I$ if $\forall x_1, x_2 \in I, x_1 < x_2 \implies f(x_1) < f(x_2)$.

   Interpretation: As you move from left to right across the interval $I$ on the x-axis (meaning $x$ is increasing), the graph of the function $y=f(x)$ is always climbing upwards. There are no flat spots or downward movements in the graph over this interval.

   Example: $f(x) = x^3$ is strictly increasing on $(-\infty, \infty)$. If $x_1 < x_2$, then $x_1^3 < x_2^3$.

2. Increasing Function (Non-Decreasing Function): A function $f$ is said to be increasing (or non-decreasing) on the interval $I$ if for any two distinct numbers $x_1$ and $x_2$ chosen from $I$, whenever $x_1$ is less than $x_2$, the corresponding function value $f(x_1)$ is less than or equal to $f(x_2)$.

   Formally: $f$ is increasing on $I$ if $\forall x_1, x_2 \in I, x_1 < x_2 \implies f(x_1) \le f(x_2)$.

   Interpretation: As you move from left to right across the interval $I$, the graph of the function is climbing upwards or staying flat. It never moves downwards. A strictly increasing function is a special case of an increasing function.

   Example: $f(x) = \begin{cases} x & , & x \le 0 \\ 0 & , & 0 < x \le 1 \\ x-1 & , & x > 1 \end{cases}$ is increasing on $(-\infty, \infty)$. It is strictly increasing on $(-\infty, 0]$ and $(1, \infty)$, and constant (thus also increasing and decreasing by the non-strict definition) on $(0, 1]$.

3. Strictly Decreasing Function: A function $f$ is said to be strictly decreasing on the interval $I$ if for any two distinct numbers $x_1$ and $x_2$ chosen from $I$, whenever $x_1$ is less than $x_2$, the corresponding function value $f(x_1)$ is strictly greater than $f(x_2)$.

   Formally: $f$ is strictly decreasing on $I$ if $\forall x_1, x_2 \in I, x_1 < x_2 \implies f(x_1) > f(x_2)$.

   Interpretation: As you move from left to right across the interval $I$, the graph of the function is always falling downwards. There are no flat spots or upward movements.

   Example: $f(x) = -x$ is strictly decreasing on $(-\infty, \infty)$. If $x_1 < x_2$, then $-x_1 > -x_2$.

4. Decreasing Function (Non-Increasing Function): A function $f$ is said to be decreasing (or non-increasing) on the interval $I$ if for any two distinct numbers $x_1$ and $x_2$ chosen from $I$, whenever $x_1$ is less than $x_2$, the corresponding function value $f(x_1)$ is greater than or equal to $f(x_2)$.

   Formally: $f$ is decreasing on $I$ if $\forall x_1, x_2 \in I, x_1 < x_2 \implies f(x_1) \ge f(x_2)$.

   Interpretation: As you move from left to right across the interval $I$, the graph of the function is falling downwards or staying flat. It never moves upwards. A strictly decreasing function is a special case of a decreasing function.

   Example: $f(x) = \begin{cases} -x & , & x \le 0 \\ 0 & , & 0 < x \le 1 \\ -(x-1) & , & x > 1 \end{cases}$ is decreasing on $(-\infty, \infty)$. It is strictly decreasing on $(-\infty, 0]$ and $(1, \infty)$, and constant (thus also increasing and decreasing by the non-strict definition) on $(0, 1]$.

5. Monotonic Function: A function $f$ is said to be monotonic on an interval $I$ if it is either increasing on $I$ or decreasing on $I$. If it is either strictly increasing or strictly decreasing on $I$, it is called strictly monotonic.

   Monotonic functions are particularly important because a strictly monotonic function on an interval has a unique inverse function over that interval.

Applications of Derivatives: Test for Monotonicity using First Derivative

The definitions of increasing and decreasing functions describe the behavior based on comparing function values. However, using the first derivative provides a much more efficient method to determine intervals of monotonicity, especially for functions defined by formulas.

The connection between the derivative and monotonicity lies in the geometric interpretation: the derivative $f'(x)$ at a point $x$ gives the slope of the tangent line to the graph of $f(x)$ at that point. Intuitively, if the tangent line at every point in an interval has a positive slope, the function must be increasing over that interval. Similarly, if the tangent line always has a negative slope, the function must be decreasing.

The formal relationship is established by the Mean Value Theorem. If $f'(x) > 0$ for all $x$ in $(a, b)$, consider any $x_1, x_2 \in [a, b]$ with $x_1 < x_2$. By the MVT, there exists a $c \in (x_1, x_2)$ such that $\frac{f(x_2) - f(x_1)}{x_2 - x_1} = f'(c)$. Since $x_1 < x_2$, $x_2 - x_1 > 0$. Also, $c \in (x_1, x_2) \subseteq (a, b)$, so $f'(c) > 0$. Thus, $\frac{f(x_2) - f(x_1)}{x_2 - x_1} > 0$. Since the denominator is positive, the numerator $f(x_2) - f(x_1)$ must be positive, meaning $f(x_2) > f(x_1)$. This proves that $f$ is strictly increasing on $[a, b]$. Similar arguments hold for the other cases.

First Derivative Test for Monotonicity

Let $f$ be a function that is continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$. The sign of the first derivative $f'(x)$ on the open interval $(a, b)$ determines the monotonicity of $f(x)$ on the entire closed interval $[a, b]$.

1. If $f'(x) > 0$ for all $x$ in the open interval $(a, b)$, then the function $f$ is strictly increasing on the closed interval $[a, b]$.

   Explanation: A positive derivative indicates a positive slope of the tangent, meaning the function is locally increasing at every point. Over an interval, this leads to a strictly increasing trend.

2. If $f'(x) < 0$ for all $x$ in the open interval $(a, b)$, then the function $f$ is strictly decreasing on the closed interval $[a, b]$.

   Explanation: A negative derivative indicates a negative slope of the tangent, meaning the function is locally decreasing at every point. Over an interval, this leads to a strictly decreasing trend.

3. If $f'(x) = 0$ for all $x$ in the open interval $(a, b)$, then the function $f$ is constant on the closed interval $[a, b]$.

   Explanation: A zero derivative indicates a zero slope of the tangent (horizontal line). If the slope is always horizontal, the function value does not change over the interval.

Important Notes and Extensions:

• The test requires differentiability on the *open* interval $(a, b)$ but the conclusion about monotonicity applies to the *closed* interval $[a, b]$, provided $f$ is continuous on $[a, b]$. Continuity ensures that the behavior on the open interval extends to include the endpoints.
• The conditions can be stated using non-strict inequalities:
  • If $f'(x) \ge 0$ for all $x$ in $(a, b)$, then $f$ is increasing (non-decreasing) on $[a, b]$. This allows for the possibility of horizontal tangent lines ($f'(x)=0$) within the interval without the function changing direction.
  • If $f'(x) \le 0$ for all $x$ in $(a, b)$, then $f$ is decreasing (non-increasing) on $[a, b]$. This allows for horizontal tangent lines within the interval.
• The points where the function's monotonicity might change are where the derivative $f'(x)$ is either equal to zero or is undefined. These points are called critical points. To find intervals of monotonicity, we typically find the critical points and the points where the function is undefined, divide the domain into intervals based on these points, and then test the sign of $f'(x)$ in each interval. This process is detailed in the next section.

The First Derivative Test for Monotonicity is a fundamental application of differentiation, enabling the detailed analysis of a function's behavior and serving as a cornerstone for finding local extrema.

Finding Intervals of Increase and Decrease

Using the First Derivative Test for Monotonicity, we can develop a systematic procedure to determine the intervals on which a function $f(x)$ is strictly increasing or strictly decreasing. This process involves finding the points where the derivative might change sign and then testing the sign of the derivative in the intervals defined by these points.

Steps for Finding Intervals of Monotonicity:

1. Determine the Domain of the Function: Identify the set of all possible input values $x$ for which $f(x)$ is defined. This is the overall region on the number line where the function exists. Consider any restrictions from square roots of negative numbers, denominators being zero, logarithms of non-positive numbers, etc.
2. Find the First Derivative: Calculate the derivative $f'(x)$ using the appropriate differentiation rules.
3. Find the Critical Points of $f(x)$: Critical points are the points in the domain of the function where the derivative is either zero or undefined. These are the only points where the function's graph can potentially change from increasing to decreasing or vice versa.
   • Set $f'(x) = 0$ and solve for $x$. The solutions are critical points where the tangent line is horizontal.
   • Find the values of $x$ where $f'(x)$ is undefined. These could occur at points where the original function has sharp corners, cusps, or vertical tangents. Remember that $x$ must be in the domain of the original function $f(x)$ to be a critical point.
4. Create Intervals on the Number Line: Use the critical points found in Step 3 (and any points where the original function $f(x)$ is discontinuous or undefined) to divide the domain of $f$ into a series of open intervals. Arrange these points in increasing order on a number line, and these points serve as boundary markers for the intervals.
5. Test the Sign of $f'(x)$ within Each Interval: Choose a single convenient test value $x_{test}$ within each open interval created in Step 4. Evaluate the sign of the derivative $f'(x_{test})$ at this test value. The sign of $f'(x)$ will be the same for all values of $x$ within that entire open interval because $f'(x)$ can only change sign at critical points or points of discontinuity.
6. State Conclusions about Monotonicity: Based on the sign of $f'(x)$ in each interval:
   • If $f'(x) > 0$ for all $x$ in an open interval, then $f(x)$ is strictly increasing on that open interval.
   • If $f'(x) < 0$ for all $x$ in an open interval, then $f(x)$ is strictly decreasing on that open interval.
7. Write the Final Answer using Closed Intervals (Optional but Common): For functions that are continuous at the critical points, the intervals of strict monotonicity are often extended to include the critical points themselves, stating where the function is "increasing" or "decreasing" (using the non-strict definitions $f'(x) \ge 0$ or $f'(x) \le 0$). If $f$ is continuous at $x=c$ where $f'(c)=0$, and the function is strictly increasing to the left of $c$ and strictly increasing to the right of $c$, then $f$ is strictly increasing over the combined interval including $c$. If it changes direction at $c$, then $c$ is included in the closed interval for increasing behavior ending at $c$, and the closed interval for decreasing behavior starting at $c$ (or vice versa). Always check the continuity of $f$ at the endpoints of the closed interval you state.

This process provides a clear roadmap for identifying where a function's graph is rising or falling.

Answer:

Increasing /Decreasing Functions (Applied Maths)

In applied mathematics, science, and engineering, functions are used to model real-world phenomena. Understanding where these functions are increasing or decreasing provides crucial insights into the behavior of the system being modeled. Analyzing the sign of the first derivative allows us to interpret the dynamic behavior of quantities such as position, velocity, cost, revenue, population size, etc.

Interpretation and Applications in Applied Contexts

The sign of the derivative $f'(x)$ tells us whether the quantity represented by $f(x)$ is growing or shrinking as $x$ increases. This has direct practical implications:

• Optimization: Finding the intervals of increase and decrease is a fundamental step in optimization problems. Local maxima occur where a function changes from increasing to decreasing (sign of $f'$ changes from positive to negative). Local minima occur where a function changes from decreasing to increasing (sign of $f'$ changes from negative to positive). These points often correspond to maximum profit, minimum cost, maximum volume, minimum surface area, etc., in real-world applications. The First Derivative Test for Local Extrema builds directly on the concept of monotonicity intervals.
• Graph Sketching and Analysis: Knowing the intervals where a function is increasing or decreasing is essential for accurately sketching its graph. It reveals the "hills" and "valleys" of the curve and its overall shape. This visual representation is often key to understanding the behavior of the modeled system.
• Economic Analysis: In business and economics, marginal analysis uses derivatives to understand rates of change.
  • If Marginal Revenue $R'(x) > 0$, it means that selling more units will increase total revenue. If $R'(x) < 0$, selling more units will decrease total revenue.
  • If Marginal Cost $C'(x) > 0$ (which is usually the case for positive production levels), total cost increases with increased production. Analyzing whether Marginal Cost is increasing ($C''(x)>0$) or decreasing ($C''(x)<0$) gives insights into economies of scale (decreasing marginal cost) or diseconomies of scale (increasing marginal cost).
  • If Marginal Profit $P'(x) > 0$, producing and selling one more unit will increase total profit. If $P'(x) < 0$, producing and selling one more unit will decrease total profit. The point of maximum profit occurs where $P'(x) = 0$ (meaning $R'(x) = C'(x)$) and $P'(x)$ changes from positive to negative (indicating the function changes from increasing profit to decreasing profit).
• Physical Models (Motion): If $s(t)$ is the position of an object at time $t$, and $v(t) = s'(t)$ is its velocity:
  • If $v(t) > 0$, the object's position is increasing, meaning it is moving in the positive direction.
  • If $v(t) < 0$, the object's position is decreasing, meaning it is moving in the negative direction.
  • If $v(t) = 0$, the object is momentarily at rest. Points where velocity changes sign indicate where the object changes direction.
  Furthermore, if $a(t) = v'(t) = s''(t) > 0$, the velocity is increasing. If $v(t) > 0$ and $a(t) > 0$, the object is speeding up. If $v(t) < 0$ and $a(t) > 0$, the object is slowing down (since velocity is becoming less negative). Similarly, if $a(t) < 0$, velocity is decreasing. If $v(t) > 0$ and $a(t) < 0$, the object is slowing down. If $v(t) < 0$ and $a(t) < 0$, the object is speeding up (since velocity is becoming more negative).
• Rate Analysis in General: Whether analyzing the rate of change of temperature, concentration of a substance, rate of flow, etc., determining if the derivative is positive or negative immediately tells us if the quantity is increasing or decreasing over time or with respect to another variable.

In applied problems, after setting up a function to model a situation, finding its derivative and analyzing its sign over the relevant domain (often restricted to physically meaningful values, like non-negative time or quantity) is a crucial step in understanding the dynamic behavior of the modeled phenomenon, such as when it's growing, shrinking, speeding up, slowing down, or reaching an optimal state.