LemonmathSelf-study sheet

Grade 12 · Lesson 1.1

Monotonicity and extrema

Name: Date: Time spent: min

Goals

Find the intervals where a function is increasing or decreasing from the sign of its derivative.
Draw a table of variations and find local maxima and minima from the change of sign of $f'$ .
Use the second derivative to recognise an extremum when that is easier.
Apply these to cubics, quartics, rational functions and a real-life problem.

Remember

$f'>0$ on $K$ means $f$ is increasing on $K$ ; $f'<0$ means decreasing. If $f'\ge0$ and it is zero at only finitely many points, $f$ is still increasing.
Steps: domain → $f'$ and its roots → table of variations → conclusion for each interval.
$f'$ changes sign $+\to-$ at $x_0$ : local maximum; $-\to+$ : local minimum. No sign change: no extremum.
Second-derivative test: $f'(x_0)=0$ and $f''(x_0)<0$ gives a local maximum, $f''(x_0)>0$ a local minimum.
Distinguish the point $x_0$ where the extremum occurs, the extreme value $f(x_0)$ , and the turning point of the graph $(x_0, f(x_0))$ .

Exercises

Exercise 1

Find the intervals on which $y=-x^3+3x^2+1$ is increasing or decreasing.

Exercise 2

Find the extrema of $y=x^4-8x^2+1$ .

Exercise 3

Determine where $y=\dfrac{x+2}{x-1}$ is increasing or decreasing.

Exercise 4

Find the extrema of $y=x+\dfrac{4}{x}$ .

Exercise 5

A function $y=f(x)$ has derivative $f'(x)=x(x-1)^2(x+2)$ for every $x\in\mathbb{R}$ . Is each claim true or false?

a) $f$ is increasing on the interval $(0, +\infty)$ .

b) $f$ has three turning points.

c) $f$ has a local maximum at $x=-2$ .

d) $f$ is decreasing on the interval $(-2, 1)$ .

Exercise 6

The function $y=x^3-3x^2-9x+4$ has a local maximum value $y_{\text{max}}$ and a local minimum value $y_{\text{min}}$ . Compute $y_{\text{max}}-y_{\text{min}}$ .

Exercise 7

The concentration of a medicine in the blood $t$ hours after an injection is $C(t)=\dfrac{30t}{t^2+9}$ (mg/l), $t>0$ . During which time interval does the concentration increase, and when does it decrease? When does it reach its local maximum, and what is that value?

Exercise 8 Review

Find the extrema of $y=x+2\cos x$ on the interval $(0, \pi)$ .

Exercise 9 Review

Prove that $y=x^3+x^2+x+1$ is increasing on $\mathbb{R}$ .

Exercise 10 Challenge

Find the intervals on which $y=x\sqrt{4-x^2}$ is increasing or decreasing, and its extrema.

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. On which of these intervals is

y=x^3-3x^2+2

decreasing?

(-\infty, 0)

.B.

(2, +\infty)

.C.

(0, 2)

.D.

(-2, 0)

Question 2. The function

y=-x^3+3x+1

has a local maximum at

x=-1

.B.

x=1

.C.

x=3

.D.

x=0

Question 3. A function

y=f(x)

has a second derivative on

\mathbb{R}

, with

f'(2)=0

and

f''(2)=-3

. Which statement is correct?

f

has a local minimum at

x=2

.B.

f

has a local maximum at

x=-3

.C.

f

has no extremum at

x=2

.D.

f

has a local maximum at

x=2

Question 4. Which function is increasing on

\mathbb{R}

y=x^3+3x-1

.B.

y=x^3-3x

.C.

y=\dfrac{2x+1}{x+3}

.D.

y=x^4+2x^2

Part II. True or false

Question 1. Let

f(x)=2x^3-3x^2-12x+1

a) $f'(x)=6x^2-6x-12$ .	T / F
b) $f$ is increasing on the interval $(-1, 2)$ .	T / F
c) The local maximum value of $f$ is $8$ .	T / F
d) The line through the two turning points of the graph has gradient $9$ .	T / F

Part III. Short answer

Question 1. Find the sum of the local maximum value and the local minimum value of

y=-x^4+8x^2-3

Question 2. The water level in a reservoir after

t

hours is modelled by

h(t)=t^3-15t^2+60t+100

(cm),

0\le t\le 12

. For how many hours does the water level fall? Give your answer to 1 decimal place.

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: if $y'>0$ on an interval, the function is increasing there; if $y'<0$ , it is decreasing.

Steps:

Compute $y'$ and factorise it.
Find the sign of $y'$ (mind the negative coefficient).

Exercise 2.

What you need: $y'$ changes sign from $-$ to $+$ at a local minimum and from $+$ to $-$ at a local maximum.

Steps:

Compute $y'=4x^3-16x$ and find its three roots.
Draw the table of variations.
Compute $y$ at these points.

Exercise 3.

What you need: $\left(\dfrac{ax+b}{cx+d}\right)'=\dfrac{ad-bc}{(cx+d)^2}$ .

Steps:

Find the domain.
Compute $y'$ and look at the sign of the numerator.

Exercise 4.

What you need: write $y'$ as one fraction to find its sign; remember $x=0$ is not in the domain.

Steps:

Compute $y'=1-\dfrac{4}{x^2}=\dfrac{x^2-4}{x^2}$ .
Draw the table of variations with the values $-2$ , $0$ , $2$ marked.

Exercise 5.

What you need: the factor $(x-1)^2$ is non-negative, so it does not make $f'$ change sign; extrema occur only where $f'$ changes sign.

Steps:

Make a sign table of $f'$ with the values $-2$ , $0$ , $1$ marked.
Read off where $f$ increases or decreases and where $f'$ changes sign.

Exercise 6.

What you need: for a cubic with two extrema and $a>0$ , the local maximum occurs at the smaller root of $y'$ .

Steps:

Solve $y'=3x^2-6x-9=0$ .
Compute $y$ at the two roots.

Exercise 7.

What you need: the quotient rule $\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}$ .

Steps:

Compute $C'(t)$ and simplify the numerator.
Find the sign of $C'(t)$ for $t>0$ .

Exercise 8.

What you need: the equation $\sin x=\dfrac12$ (basic trigonometric equations, grade 11) and the second-derivative test.

Steps:

Solve $y'=1-2\sin x=0$ on $(0, \pi)$ .
Compute $y''=-2\cos x$ at each root.

Exercise 9.

What you need: a quadratic expression with $\Delta<0$ always has the sign of $a$ (see the sign of a quadratic expression).

Steps:

Compute $y'$ .
Compute the discriminant of $y'$ and look at the sign of $a$ .

Exercise 10.

What you need: $\left(\sqrt{u}\right)'=\dfrac{u'}{2\sqrt{u}}$ ; a square root is defined only when the expression under it is non-negative.

Steps:

Find the domain.
Compute $y'$ and write it as a single fraction.
Find the sign of the numerator on $(-2, 2)$ .

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