Increasing and decreasing functions

A function is increasing on an interval if larger inputs give larger outputs there; decreasing if larger inputs give smaller outputs. The derivative tells us instantly: $f' > 0$ means increasing, $f' < 0$ means decreasing. This single observation underpins all the optimisation in this chapter.

Definitions

$f$ is increasing on $I$ if for all $x_1, x_2 \in I$, $x_1 < x_2 \Rightarrow f(x_1) \le f(x_2)$.

$f$ is strictly increasing if $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$.

Similarly for decreasing and strictly decreasing, with $\le$ replaced by $\ge$ (or $<$ by $>$).

The monotonicity criterion

Theorem. Let $f$ be differentiable on an open interval $I$.

• $f' \ge 0$ on $I \Rightarrow f$ is increasing on $I$.
• $f' > 0$ on $I$ (except possibly at finitely many points) $\Rightarrow f$ is strictly increasing.
• $f' \le 0$ on $I \Rightarrow f$ is decreasing.
• $f' < 0$ on $I$ (except possibly at finitely many points) $\Rightarrow f$ is strictly decreasing.
• $f' = 0$ on $I \Rightarrow f$ is constant.

Proof. Apply MVT: for $x_1 < x_2$, $f(x_2) - f(x_1) = f'(c)(x_2 - x_1)$ with $c$ between. The sign of $f(x_2) - f(x_1)$ matches that of $f'(c)$.

The procedure for finding intervals of monotonicity

1. Compute $f'(x)$.
2. Solve $f'(x) = 0$ to find critical points.
3. Solve $f'(x)$ undefined (vertical tangents, corners) for additional critical points.
4. Draw a sign chart: mark critical points on a number line and check the sign of $f'$ in each subinterval.
5. Read off the intervals of monotonicity.

Worked example

$f(x) = x^3 - 3x + 2$. $f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1)$.

Zeros at $x = -1, 1$. Sign chart:

• $x < -1$: $f'(x) > 0$.
• $-1 < x < 1$: $f'(x) < 0$.
• $x > 1$: $f'(x) > 0$.

So $f$ is increasing on $(-\infty, -1) \cup (1, \infty)$ and decreasing on $(-1, 1)$.

Worked examples

Example 1. Find the intervals where $f(x) = 2x^3 - 9 x^2 + 12 x + 5$ is increasing.

$f'(x) = 6x^2 - 18 x + 12 = 6(x^2 - 3x + 2) = 6(x - 1)(x - 2)$.

Sign chart: $f' > 0$ for $x < 1$ or $x > 2$; $f' < 0$ for $1 < x < 2$.

Increasing on $(-\infty, 1) \cup (2, \infty)$; decreasing on $(1, 2)$.

Example 2. Show that $f(x) = \tan x - x$ is increasing on $(0, \pi/2)$.

$f'(x) = \sec^2 x - 1 = \tan^2 x \ge 0$, with equality only at $x = 0$. So $f$ is increasing.

Example 3. Find the values of $x$ for which $f(x) = x e^x$ is decreasing.

$f'(x) = e^x + x e^x = e^x(1 + x)$. $e^x > 0$, so $f'(x) < 0$ iff $1 + x < 0$, i.e., $x < -1$. Decreasing on $(-\infty, -1)$.

Example 4. Show $f(x) = \log x / x$ is decreasing for $x > e$.

$f'(x) = \dfrac{1/x \cdot x - \log x \cdot 1}{x^2} = \dfrac{1 - \log x}{x^2}$. For $x > e$, $\log x > 1$, so $f'(x) < 0$. Decreasing.

Example 5. For what values of $a$ is $f(x) = (a + 2) x^3 - 3 a x^2 + 9 a x - 1$ decreasing for all real $x$?

$f'(x) = 3(a + 2) x^2 - 6 a x + 9 a$. For $f$ to be decreasing for all $x$, need $f'(x) \le 0$ for all $x$, with equality only at finitely many points.

This requires $a + 2 < 0$ (leading coefficient negative , so the parabola opens down) AND discriminant $\le 0$ (so $f'$ never crosses zero, except possibly at one point).

Discriminant: $36 a^2 - 4 \cdot 3(a + 2) \cdot 9 a = 36 a^2 - 108 a (a + 2) = 36 a^2 - 108 a^2 - 216 a = -72 a^2 - 216 a$. Wait, that should be $\le 0$:

$-72 a^2 - 216 a \le 0 \Rightarrow 72 a^2 + 216 a \ge 0 \Rightarrow 72 a (a + 3) \ge 0 \Rightarrow a \le -3$ or $a \ge 0$.

Combined with $a + 2 < 0$ (i.e., $a < -2$): $a \le -3$.

Example 6. Show that the function $f(x) = \sin x + \cos x$ is increasing on $(0, \pi/4)$.

$f'(x) = \cos x - \sin x = \sqrt 2 \cos(x + \pi/4)$. On $(0, \pi/4)$, $x + \pi/4 \in (\pi/4, \pi/2)$, where cosine is positive. So $f' > 0$, increasing.

Try it yourself

1. Find the intervals of increase/decrease for $f(x) = x^2 - 4x + 3$.
2. Find the intervals of increase/decrease for $f(x) = x^3 + 3x^2 - 9x + 5$.
3. Show $f(x) = e^x$ is strictly increasing on $\mathbb{R}$.
4. Show $f(x) = \sin x$ is strictly increasing on $(-\pi/2, \pi/2)$.
5. Find the intervals where $f(x) = (x + 1)^3 (x - 3)^3$ is decreasing.
6. Show $f(x) = \log(1 + x) - x/(1 + x)$ is increasing on $(0, \infty)$.
7. Show $f(x) = x - \sin x$ is increasing on $\mathbb{R}$.
8. Determine the intervals where $f(x) = \tan^{-1}(\sin x + \cos x)$ is decreasing.
9. Find $a$ such that $f(x) = x^3 + a x^2 + 3 x$ is strictly increasing on $\mathbb{R}$.
10. Show that $f(x) = 4 \sin x - 3 \sin x \cos x + 5x$ is increasing on $\mathbb{R}$.
11. Find the intervals of monotonicity for $f(x) = \log x - x$.
12. Find the intervals where $f(x) = x + 4/x$ is decreasing.
13. Show $f(x) = \cos x$ is decreasing on $(0, \pi)$.
14. Find all $x$ where $f(x) = x^3 - 6x^2 + 9x + 15$ is increasing.

Pitfalls / Tricks

• $f' > 0$ ⇒ strictly increasing, but $f' \ge 0$ allows $f$ constant on subintervals.
• Always include the critical points correctly on the sign chart.
• If $f'$ is undefined somewhere in the domain, that point is also critical.
• For polynomials in factored form, sign chart is automatic , just check signs of each factor.
• Monotonic + continuous ⇒ injective. Useful in proving existence of inverses.

Next, tangents and normals.