Understanding Monotonicity and Extremum of Functions

Monotonicity and extremum of functions are foundational ideas in calculus concerning the behaviour of real-valued functions under differentiation. This topic studies how the sign of the derivative determines intervals where a function is consistently increasing or decreasing, and precisely characterizes points where a function attains its local maximum or minimum values.

Definition of Monotonic Increasing and Decreasing Functions

A function $f(x)$ is said to be monotonic increasing on an interval $I$ if for every $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) \leq f(x_2)$. In the strictly increasing case, the inequality is always strict: $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.

A function $f(x)$ is monotonic decreasing on $I$ if for all $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) \geq f(x_2)$. It is strictly decreasing if $f(x_1) > f(x_2)$ for all $x_1 < x_2$ in $I$.

A function that is monotonic (either increasing or decreasing) throughout an interval is called monotonic on that interval. If it never changes the direction of increase or decrease, it is said to be monotonic over its domain.

Criterion for Monotonicity Using Derivatives

Let $f$ be a function differentiable on an open interval $I$. If $f'(x) \geq 0$ for all $x \in I$, then $f(x)$ is monotonic increasing on $I$.

If $f'(x) \leq 0$ for all $x \in I$, then $f(x)$ is monotonic decreasing on $I$.

If $f'(x) > 0$ for all $x \in I$, then $f(x)$ is strictly increasing on $I$; if $f'(x) < 0$ for all $x \in I$, $f(x)$ is strictly decreasing.

If $f'(x) = 0$ everywhere on $I$, then $f(x)$ is constant on $I$.

If $f'(x)$ takes both positive and negative values on $I$, $f(x)$ is not monotonic on $I$. For more detail on practical tests for monotonicity, refer to Increasing And Decreasing Functions.

Monotonicity of Common Functions

For $f(x) = x^r$ with $r > 0$, $f'(x) = r x^{r-1}$. On $(0, \infty)$, $f'(x) > 0$, so the function is strictly increasing there. For $r < 0$, $f(x)$ is strictly decreasing on $(0, \infty)$.

The exponential function $f(x) = e^x$ has $f'(x) = e^x > 0$ for all $x \in \mathbb{R}$, so it is strictly increasing everywhere. For a detailed understanding and further examples, see Derivative Examples.

Definition of Extremum: Local Maxima and Minima

An interior point $c$ of the domain of $f$ is a point of local (or relative) maximum if there exists $\delta > 0$ such that $f(x) \leq f(c)$ for all $x$ in the open interval $(c - \delta, c + \delta)$, $x \ne c$.

An interior point $c$ of the domain is a point of local (or relative) minimum if there exists $\delta > 0$ such that $f(x) \geq f(c)$ for all $x$ in $(c - \delta, c + \delta)$, $x \ne c$.

A local extremum refers to either a local maximum or minimum. Such points are also called stationary points if $f'(c) = 0$ exists.

Necessary and Sufficient Conditions for Extreme Values

If $f$ is differentiable at $c$ and $f$ achieves a local extremum at $c$, then necessarily $f'(c) = 0$.

This is a necessary but not sufficient condition. Points where $f'(c) = 0$ are called critical points. To determine if a given critical point is a maximum, minimum, or neither, further analysis via the first or second derivative test is required.

First Derivative Test for Local Extrema

Let $c$ be a critical point. If $f'(x)$ changes sign from positive to negative as $x$ increases through $c$, then $f$ has a local maximum at $c$.

If $f'(x)$ changes sign from negative to positive as $x$ increases through $c$, then $f$ has a local minimum at $c$.

If $f'(x)$ does not change sign around $c$, then $f(c)$ is neither a local maximum nor a local minimum.

Second Derivative Test for Local Extrema

Let $f'(c) = 0$ and suppose $f''$ exists at $c$.

If $f''(c) > 0$, then $f$ has a local minimum at $c$.

If $f''(c) < 0$, then $f$ has a local maximum at $c$.

If $f''(c) = 0$, the test is inconclusive. Exam Tip: Always revert to the first derivative test if the second derivative test fails.

Extreme Values on Closed Intervals

If $f$ is continuous on $[a, b]$, then its maximum and minimum values on $[a, b]$ are achieved either at critical points inside $(a, b)$ where $f'(x) = 0$ or $f'(x)$ is undefined, or at the endpoints $a$ and $b$.

To find the greatest and least values on $[a, b]$, calculate $f(x)$ at all such points and compare. This procedure is especially important in polynomial case studies such as in Quadratic Polynomial Max And Min Values.

Interrelation of Monotonicity and Invertibility

If $f$ is strictly monotonic (strictly increasing or strictly decreasing) on an interval $I$, then $f$ is invertible on $I$, and its inverse function is also strictly monotonic with the same sense as $f$.

Exam-Focused Example: Testing Monotonicity

Example: Let $f(x) = x^3 - 12x^2 + 36x + 17$ on $[1, 10]$. Find its maximum value.

First, compute $f'(x)$:

$f'(x) = \dfrac{d}{dx}(x^3 - 12x^2 + 36x + 17) = 3x^2 - 24x + 36$

Set $f'(x) = 0$ to find critical points:

$3x^2 - 24x + 36 = 0$

Divide both sides by $3$:

$x^2 - 8x + 12 = 0$

Factor the quadratic:

$(x - 6)(x - 2) = 0$

So the critical points are $x = 2$ and $x = 6$.

Calculate $f(x)$ at $x = 1, 2, 6, 10$:

$f(1) = 1 - 12 + 36 + 17 = 42$

$f(2) = 8 - 48 + 72 + 17 = 49$

$f(6) = 216 - 432 + 216 + 17 = 17$

$f(10) = 1000 - 1200 + 360 + 17 = 177$

Comparing these values, the greatest is $f(10) = 177$.

Solution: The maximum value of $f(x)$ on $[1, 10]$ is $177$ at $x = 10$.

Characterization of Points of Inflection

A point $c$ is a point of inflection of $f$ if $f$ is continuous at $c$, $f''(c) = 0$, and $f''(x)$ changes sign as $x$ passes through $c$. At such points, the concavity of the curve changes but it is not a maximum or minimum.

Key Results and Exam Cues in Monotonicity and Extremum

Between consecutive local maxima (or minima) of a continuous differentiable function, there must occur a local minimum (or maximum) respectively.

If for $x \in (a, b)$, $f'(x)$ retains the same sign except at isolated points, $f(x)$ is monotonic on $(a, b)$. For detailed examples, refer to Higher Order Derivative Examples.

If $f$ and $g$ are both strictly increasing (or both strictly decreasing) on $I$ and $g \circ f$ is defined, then $g \circ f$ is strictly increasing on $I$.

If $f$ is strictly increasing and $g$ is strictly decreasing on $I$, then $g \circ f$ is strictly decreasing on $I$.

Common Errors in Analysis of Monotonicity and Extremum

Do not conclude monotonicity solely from $f'(x) \geq 0$ or $f'(x) \leq 0$ without verifying the sign is nonzero on a nontrivial interval, otherwise $f$ may just be constant.

An extremum may exist at isolated points where $f'(x)$ does not exist. Always check non-differentiable points separately.

FAQs: Standard Results on Maximum and Minimum Values

The rectangle of maximal area for a fixed perimeter is a square. For a fixed area, the rectangle of minimal perimeter is also a square. The largest rectangle inscribed in a circle is a square, and the largest area triangle inscribed in a circle is equilateral.

The maximum and minimum values of a continuous function on a closed interval $[a, b]$ always occur either at endpoints or at stationary points in $(a, b)$. The local maximum at a point need not be the absolute maximum on the interval. For a broader context on derivatives related to monotonicity, see Partial Derivative Of Functions.