Where a function increases and decreases, and extremum points

Algebra and calculus, grade 10 (4 units) Lesson 13 of 20Learning

Where a function increases and decreases, and extremum points

The sign of the derivative

The derivative at a point is the slope of the tangent, so its sign tells which way the graph goes there:

$\displaystyle f'(x)>0$ – the tangent rises, and the function increases.
$\displaystyle f'(x)<0$ – the tangent falls, and the function decreases.
$\displaystyle {f'(x)}={0}$ – the tangent is horizontal.

Where the function increases is the solution of the inequality $\displaystyle f'(x)>0$ , and where it decreases – of $\displaystyle f'(x)<0$ .

A point that may be an extremum

At an extremum point the function stops increasing and starts decreasing (a maximum), or the other way round (a minimum). There the tangent is horizontal, so the extremum points are among the solutions of $\displaystyle {f'(x)}={0}$ .

\displaystyle f'(x)

\displaystyle =

\displaystyle 0

A solution of $\displaystyle {f'(x)}={0}$ is a point that may be an extremum.
A table of signs decides: check the sign of $\displaystyle f'$ on both sides of each such point.
From + to −: a maximum point. From − to +: a minimum point.

Example:

\displaystyle {f(x)}={x^3-3x}

The example below finds where $\displaystyle {f(x)}={x^3-3x}$ increases and decreases, and its extremum points.

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

The derivative is 0 at $\displaystyle {x}={-1}$ and at $\displaystyle {x}={1}$ . We substitute one number from each interval in the derivative and get the table:

$\displaystyle x$	$\displaystyle x<-1$	$\displaystyle -1$	$\displaystyle -1<x<1$	$\displaystyle 1$	$\displaystyle x>1$
$\displaystyle f'(x)$	+	0	−	0	+
$\displaystyle f(x)$	increases	maximum	decreases	minimum	increases

\displaystyle f(x)=x^{3}-3x

\displaystyle A=(-1,2)

\displaystyle B=(1,-2)

The function increases where $\displaystyle x<-1$ and where $\displaystyle x>1$ , and decreases where $\displaystyle -1<x<1$ .
$\displaystyle {f(-1)}={2}$ : a maximum point $\displaystyle (-1,2)$ .
$\displaystyle {f(1)}={-2}$ : a minimum point $\displaystyle (1,-2)$ .

A derivative of 0 with no extremum

Not every point where $\displaystyle {f'(x)}={0}$ is an extremum point. For $\displaystyle {f(x)}={x^3}$ the derivative is $\displaystyle {f'(x)}={3x^2}$ :

$\displaystyle {f'(0)}={0}$ : the tangent at the origin is horizontal.
On both sides of 0 the derivative is positive, so the function increases before 0 and after it.
There is no extremum at the origin – the table is what decides.

\displaystyle f(x)=x^{3}

Local and absolute extremum, and an extremum at the end

An extremum point is local when its value is the greatest (or the least) only close around it. An absolute value is the greatest (or the least) in the whole domain.

$\displaystyle x^3-3x$ , defined for every $\displaystyle x$ , has no absolute maximum: as $\displaystyle x$ grows, $\displaystyle f(x)$ grows without end.
When the domain is closed, say $\displaystyle 0\leq x\leq3$ , there is also an extremum at the end of the domain.

The example below finds the extremum points of $\displaystyle {f(x)}={x^3-3x}$ in the domain $\displaystyle 0\leq x\leq3$ :

$\displaystyle {f(0)}={0}$ : from 0 to 1 the function decreases, so $\displaystyle (0,0)$ is a maximum point at the end.
$\displaystyle {f(1)}={-2}$ : a minimum point, and also the absolute minimum.
$\displaystyle {f(3)}={18}$ : from 1 to 3 the function increases, so $\displaystyle (3,18)$ is a maximum point at the end, and also the absolute maximum.

Exercise 1

The function is $\displaystyle {f(x)}={2x^3-3x^2-12x}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=2x^{3}-3x^{2}-12x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Solve the inequality $\displaystyle f'(x)>0$ : where does the function increase?

Mark the extremum points of the function on the graph.

\displaystyle f(x)=2x^{3}-3x^{2}-12x

Write where the function decreases.
Say the kind of each extremum point.

Exercise 2

The function is $\displaystyle {f(x)}={-x^3+3x^2}$ .

Write an inequality and solve it: where does the function increase?

Mark the extremum points of the function on the graph.

\displaystyle f(x)=-x^{3}+3x^{2}

Say the kind of each extremum point.
Give the reason from where the function increases.

Exercise 3

The function is $\displaystyle {f(x)}={x^4-2x^2}$ .

Write the equation $\displaystyle {f'(x)}={0}$ and solve it.

Make a table of signs of $\displaystyle f'(x)$ .
Write where the function increases and decreases, and the kind of each extremum point.

Mark the extremum points on the graph.

\displaystyle f(x)=x^{4}-2x^{2}

Exercise 4

The function is $\displaystyle {f(x)}={8(2x-1)^3}$ from Exercise 4 of the last lesson.
Its derivative is $\displaystyle {f'(x)}={48(2x-1)^2}$ .

Solve the inequality $\displaystyle f'(x)>0$ .

At $\displaystyle {x}={\frac{1}{2}}$ the derivative is 0.
Is it an extremum point?
Does the function decrease anywhere?
Give your reasons.

Exercise 5

The function $\displaystyle {f(x)}={x^3-12x}$ is defined in the domain $\displaystyle -3\leq x\leq5$ .

Solve the equation $\displaystyle {f'(x)}={0}$ .

Work out the values of the function at the ends of the domain and at the points you found.

$\displaystyle f(x)=x^{3}-12x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

$\displaystyle f(x)=x^{3}-12x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

$\displaystyle f(x)=x^{3}-12x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

$\displaystyle f(x)=x^{3}-12x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write all the extremum points of the function in the domain, the extremum points at the ends included.
What is the absolute maximum, and what is the absolute minimum?

Formulas: algebra and calculus, grade 10 (4 units)

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Algebraic technique

The square of a sum
$\displaystyle (a+b)^2$ $\displaystyle =$ $\displaystyle a^2+2ab+b^2$
The square of a difference
$\displaystyle (a-b)^2$ $\displaystyle =$ $\displaystyle a^2-2ab+b^2$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$
The roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units to the right ( $\displaystyle p<0$ – to the left)
$\displaystyle y=f(x-p)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A stretch ( $\displaystyle |a|>1$ ) or a shrink ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The derivative of a product
$\displaystyle (f(x)\cdot g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$