f is strictly increasing on R. Which describes g(x) = f(-x)?

Increasing on one part of R and decreasing on another

Increasing and Decreasing Functions for CUET Maths

Q: What is an increasing function?

A function f is strictly increasing on an interval I when a larger input always gives a larger output: for x₁ f(x₂).

Q: How do you find the intervals where a function is increasing?

Solve f'(x) = 0 to find where the sign can change, and mark those points, and any point where f is not defined, on the number line. They cut the line into intervals. In each interval, the sign of f' at any one test point is its sign throughout: positive means increasing, negative means decreasing.

Q: Where are sine and cosine increasing?

sin x is strictly increasing on (-π/2, π/2), where its derivative cos x is positive, and strictly decreasing on (π/2, 3π/2). cos x is strictly decreasing on (0, π), where its derivative -sin x is negative, and strictly increasing on (π, 2π). The pattern repeats every 2π.

Q: How do you find k so that a function is increasing on R?

Require f'(x) ≥ 0 for every real x and turn that into a condition on k. When f'(x) is a quadratic ax^2 + bx + c, it stays ≥ 0 exactly when a > 0 and the discriminant b^2 - 4ac ≤ 0.

About 5 min to learn · 74 questions

Increasing and decreasing functions are part of Section A of CUET Maths, the section every candidate answers, whether you take Mathematics or Applied Mathematics. One idea does all the work: the sign of the derivative tells you where a function rises and where it falls, and the same idea finds maxima and minima in the next chapter. Learn each rule here, from the definition to finding the intervals and the conditions on a constant, with worked examples, time-saving shortcuts and the traps CUET sets, then practise on free questions in CUET's own formats, each fully solved.

Syllabus area: Section A1

On this page · 10 sections

01What increasing and decreasing mean
02The derivative test
03Finding the intervals
04Trigonometric functions
05Conditions on a constant
06How CUET asks this chapter
07Exam shortcuts
08Common mistakes
09Quick revision
10Check your understanding

What increasing and decreasing mean

The ball from the last chapter, at height $h = 20t - 5t^2$ metres after $t$ seconds, climbs for 2 seconds and then falls back, landing at $t = 4$ . On the interval $(0, 2)$ a later time always means a greater height; on $(2, 4)$ it always means a smaller one. Those two sentences are exactly the definitions.

What is an increasing function?

A function $f$ is strictly increasing on an interval $I$ when a larger input always gives a larger output: for $x_1 < x_2$ in $I$ , $f(x_1) < f(x_2)$ . It is strictly decreasing when a larger input always gives a smaller output: $x_1 < x_2$ gives $f(x_1) > f(x_2)$ .

With $\le$ in place of $<$ , the function is just increasing, and it is allowed to stay level for a while; with $\ge$ , it is just decreasing. So every strictly increasing function is increasing, but a constant function is increasing without being strictly increasing.

Example 1 · From the definition

Show that $f(x) = 2x + 9$ is strictly increasing on $\mathbb{R}$ .

Take any $x_1 < x_2$ . Then $2x_1 < 2x_2$ , and adding 9 keeps the order: $2x_1 + 9 < 2x_2 + 9$ .
That is $f(x_1) < f(x_2)$ for every such pair.

Answer

$f$ is strictly increasing on $\mathbb{R}$ . The slope, 2, decides; the constant 9 only moves the line up.

The same check shows that $f(x) = \frac{1}{x}$ is strictly decreasing on $(0, \infty)$ : for $0 < x_1 < x_2$ , $\frac{1}{x_1} > \frac{1}{x_2}$ . A function can also be increasing or decreasing at a point $x_0$ , which means it is so on some small open interval around $x_0$ .

The derivative test

Checking every pair of points is slow. The derivative is the slope of the graph, and a graph that slopes upwards everywhere on an interval is rising there.

How do you check whether a function is increasing or decreasing?

Find its derivative and look at the sign on the interval. If $f'(x) > 0$ at every point of an interval, $f$ is strictly increasing there; if $f'(x) < 0$ at every point, it is strictly decreasing; and if $f'(x) = 0$ at every point, it is constant.

Example 2 · Increasing everywhere

Show that $f(x) = x^3 + 5x - 1$ is strictly increasing on $\mathbb{R}$ .

$f'(x) = 3x^2 + 5$ .
$3x^2 \ge 0$ for every $x$ , so $f'(x) \ge 5 > 0$ .

Answer

$f'(x) > 0$ everywhere, so $f$ is strictly increasing on $\mathbb{R}$ .

Watch out

The test does not work backwards. $f(x) = x^5$ is strictly increasing on $\mathbb{R}$ , yet $f'(0) = 0$ . A derivative that touches 0 at single points, with the same sign on both sides, does not stop a function from strictly increasing. So "strictly increasing" does not mean " $f'(x) > 0$ everywhere".

Finding the intervals

How do you find the intervals where a function is increasing?

Solve $f'(x) = 0$ to find where the sign can change, and mark those points, and any point where $f$ is not defined, on the number line. They cut the line into intervals. In each interval, the sign of $f'$ at any one test point is its sign throughout: positive means increasing, negative means decreasing.

Example 3 · A cubic

Find the intervals on which $f(x) = 2x^3 - 9x^2 + 12x + 1$ is strictly increasing and strictly decreasing.

$f'(x) = 6x^2 - 18x + 12 = 6(x - 1)(x - 2)$ , which is 0 at $x = 1$ and $x = 2$ .
Test the sign in each piece:

Interval	Test point	Sign of $f'(x)$	$f$ is
$(-\infty, 1)$	$x = 0$	$6(-1)(-2) > 0$	increasing
$(1, 2)$	$x = 1.5$	$6(0.5)(-0.5) < 0$	decreasing
$(2, \infty)$	$x = 3$	$6(2)(1) > 0$	increasing

Answer

Strictly increasing on $(-\infty, 1) \cup (2, \infty)$ and strictly decreasing on $(1, 2)$ .

Watch out

A point where the function is not defined splits the line too. $f(x) = x + \frac{9}{x}$ has $f'(x) = \frac{x^2 - 9}{x^2}$ , which is negative for $0 < \lvert x \rvert < 3$ , so $f$ decreases on $(-3, 0) \cup (0, 3)$ , never across $x = 0$ .

A squared factor in $f'(x)$ , such as $(x - 2)^2$ , is 0 at its root but does not change sign there, so it does not split an interval of increase or decrease.

Trigonometric functions

Where are sine and cosine increasing?

$\sin x$ is strictly increasing on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ , where its derivative $\cos x$ is positive, and strictly decreasing on $\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$ . $\cos x$ is strictly decreasing on $(0, \pi)$ , where its derivative $-\sin x$ is negative, and strictly increasing on $(\pi, 2\pi)$ . The pattern repeats every $2\pi$ .

For a multiple angle, the intervals shrink: $\cos kx$ decreases while $kx$ runs from 0 to $\pi$ , that is on $\left(0, \frac{\pi}{k}\right)$ .

Example 4 · A double angle

On $(0, \pi)$ , where is $f(x) = \cos 2x$ increasing and where is it decreasing?

$f'(x) = -2\sin 2x$ .
For $x$ in $\left(0, \frac{\pi}{2}\right)$ , $2x$ is in $(0, \pi)$ and $\sin 2x > 0$ , so $f'(x) < 0$ . For $x$ in $\left(\frac{\pi}{2}, \pi\right)$ , $2x$ is in $(\pi, 2\pi)$ and $\sin 2x < 0$ , so $f'(x) > 0$ .

Answer

Decreasing on $\left(0, \frac{\pi}{2}\right)$ and increasing on $\left(\frac{\pi}{2}, \pi\right)$ .

For a sum such as $\sin x + \cos x$ , write the derivative as a single sine or cosine first: $\cos x - \sin x = \sqrt{2}\cos\left(x + \frac{\pi}{4}\right)$ , and read its sign from that.

Conditions on a constant

How do you find k so that a function is increasing on R?

Require $f'(x) \ge 0$ for every real $x$ and turn that into a condition on $k$ . When $f'(x)$ is a quadratic $ax^2 + bx + c$ , it stays $\ge 0$ exactly when $a > 0$ and the discriminant $b^2 - 4ac \le 0$ .

Example 5 · A cubic that never falls

Find the values of $k$ for which $f(x) = x^3 - 6x^2 + kx + 1$ is increasing on $\mathbb{R}$ .

$f'(x) = 3x^2 - 12x + k$ , an upward parabola.
It stays $\ge 0$ when its discriminant is $\le 0$ : $144 - 12k \le 0$ .

Answer

k \ge 12

On a smaller interval, the condition has to hold at the point of the interval where $f'$ is least. For $f(x) = x^2 + ax$ to increase on $(1, 4)$ , $f'(x) = 2x + a$ must be $\ge 0$ from $x = 1$ on, which gives $a \ge -2$ . With a trigonometric term, use its extreme value: $kx + \cos x$ increases on $\mathbb{R}$ when $k - \sin x \ge 0$ for every $x$ , that is $k \ge 1$ .

How CUET asks this chapter

Definitions: what strictly increasing means, or what $f(a) > f(b)$ says about $a$ and $b$ for a decreasing $f$ , often as a statement question.
Whole line: whether a function such as $x^3 + 2x$ or $2x + \cos x$ is increasing or decreasing on $\mathbb{R}$ .
Intervals: where a quadratic, cubic, quartic, exponential or logarithmic function increases or decreases.
Trigonometric: the interval on which $\sin x$ , $\cos x$ or a multiple angle rises or falls, or which of several functions falls on a given interval.
Constants: the values or least value of $k$ that keep a function increasing.

Exam shortcuts

$f'(x) > 0$ on an interval: strictly increasing there; $f'(x) < 0$ : strictly decreasing.
A linear function's slope alone decides; the constant term never does.
Sign of a factored $f'$ : a squared factor never changes sign.
$\cos kx$ decreases on $\left(0, \frac{\pi}{k}\right)$ .
$f'$ a quadratic: $\ge 0$ for all $x$ when $a > 0$ and $b^2 - 4ac \le 0$ .
$e^x$ , $\log x$ and $x^3$ increase; $e^{-x}$ decreases.

Common mistakes

Reading the sign of $f'$ the wrong way round and giving the decreasing interval.
Splitting an interval at the zero of a squared factor.
Joining intervals across a point where the function is not defined.
Thinking a strictly increasing function must have $f'(x) > 0$ at every point.
Using the wrong end of an interval in a condition on a constant.

Quick revision

Definitions

Strictly increasing: $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$
Increasing allows $\le$ : a constant function is increasing

Derivative test

$f' > 0$ : increasing; $f' < 0$ : decreasing; $f' = 0$ on an interval: constant
Isolated zeros of $f'$ do not spoil strict increase

Method

Solve $f'(x) = 0$ , split the line, test one point per piece
$\sin x$ rises on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ ; $\cos x$ falls on $(0, \pi)$
For all $x$ : $a > 0$ and $b^2 - 4ac \le 0$ in $f'$

Where a function stops rising and starts falling, it reaches a peak: maxima and minima.

Next · 09Quick revision

Check your understanding

74 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.

Increasing and decreasing on an intervalQ1 of 74Q1

Increasing and decreasing on an interval

11 questions

Check your understanding

Increasing and decreasing on an interval

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The derivative test

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Finding the intervals

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Trigonometric functions

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Conditions on a constant

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