Where a function rises and falls, and extremum points

Calculus, grade 10 (5 units) Lesson 12 of 26Learning

Where a function rises and falls, and extremum points

The sign of the derivative

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

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

\displaystyle f'(x)>0

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

A point that may be an extremum

At a maximum point the function stops rising and starts falling, and at a minimum point – the other way round. At both the tangent is horizontal, so every extremum point is a solution of the equation:

\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 $\displaystyle f'$ in each interval between these points.
$\displaystyle f'$ goes from + to −: a maximum point. From − to +: a minimum point. The same sign on both sides: no extremum.

Example:

\displaystyle {f(x)}={x^3-6x^2+9x}

$\displaystyle {f(x)}={x^3-6x^2+9x}={x(x-3)^2}$ is the function from the first lesson. There we read its extremum points off the graph; the example below finds them by working them out.

f'(x)=3x^2-12x+9=3(x-1)(x-3)

The derivative is 0 at $\displaystyle {x}={1}$ and at $\displaystyle {x}={3}$ . We substitute one number from each interval in the derivative – say $\displaystyle {f'(0)}={9}$ , $\displaystyle {f'(2)}={-3}$ , $\displaystyle {f'(4)}={9}$ – and get the table:

$\displaystyle x$	$\displaystyle x<1$	$\displaystyle 1$	$\displaystyle 1<x<3$	$\displaystyle 3$	$\displaystyle x>3$
$\displaystyle f'(x)$	+	0	−	0	+
$\displaystyle f(x)$	rises	maximum	falls	minimum	rises

\displaystyle f(x)=x^{3}-6x^{2}+9x

\displaystyle A=(1,4)

\displaystyle B=(3,0)

The function rises where $\displaystyle x<1$ and where $\displaystyle x>3$ , and falls where $\displaystyle 1<x<3$ .
$\displaystyle {f(1)}={1-6+9}={4}$ : a maximum point $\displaystyle (1,4)$ .
$\displaystyle {f(3)}={27-54+27}={0}$ : a minimum point $\displaystyle (3,0)$ .

A derivative of 0 with no extremum

Not every point where $\displaystyle {f'(x)}={0}$ is an extremum point. The example below checks $\displaystyle {f(x)}={x^3-3x^2+3x}$ :

$\displaystyle {f'(x)}={3x^2-6x+3}={3(x-1)^2}$ , which is 0 only at $\displaystyle {x}={1}$ .
On both sides of 1 the derivative is positive: $\displaystyle (x-1)^2>0$ .
So the function rises before 1 and after it too. At the point $\displaystyle (1,1)$ the tangent is horizontal, but there is no extremum.

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

\displaystyle A=(1,1)

Local and absolute extremum, and an extremum at the end

An extremum point is local: its value is the greatest (or the least) only close around it. The greatest value in the whole domain is the absolute maximum, and the least – the absolute minimum.

$\displaystyle x^3-6x^2+9x$ , 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 $\displaystyle 0\leq x\leq5$ , there is also an extremum at the end at each end of the interval.

The example below finds the extremum points of $\displaystyle {f(x)}={x^3-6x^2+9x}$ in the interval $\displaystyle 0\leq x\leq5$ . We work out the values at the ends and at the points that may be extrema:

$\displaystyle {f(0)}={0}$ : from 0 to 1 the function rises, so $\displaystyle (0,0)$ is a minimum point at the end.
$\displaystyle {f(1)}={4}$ : a local maximum point.
$\displaystyle {f(3)}={0}$ : a local minimum point.
$\displaystyle {f(5)}={125-150+45}={20}$ : from 3 to 5 the function rises, so $\displaystyle (5,20)$ is a maximum point at the end.

The absolute maximum in the interval is 20, at $\displaystyle {x}={5}$ . The absolute minimum is 0, reached twice: at $\displaystyle {x}={0}$ and at $\displaystyle {x}={3}$ .

Exercise 1

The function is $\displaystyle {f(x)}={x^3-3x^2-9x+5}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x^{3}-3x^{2}-9x+5$

/ – 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 $\displaystyle f$ rise?

Mark the extremum points of $\displaystyle f$ on the graph.

\displaystyle f(x)=x^{3}-3x^{2}-9x+5

Write where $\displaystyle f$ falls.
Write the extremum points and the kind of each.

Exercise 2

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

Write an inequality and solve it: where does $\displaystyle f$ rise?

Mark the extremum points of $\displaystyle f$ on the graph.

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

Decide the kind of each extremum point.
Explain by where the function rises, as you found.

Exercise 3

The function is $\displaystyle {f(x)}={x^4-8x^2+3}$ .

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

Build a table of signs of $\displaystyle f'(x)$ .
Write where the function rises and falls, and its extremum points and their kinds.

Mark the extremum points on the graph.

\displaystyle f(x)=x^{4}-8x^{2}+3

Exercise 4

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

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

Does $\displaystyle f$ have extremum points?
Is there anywhere it falls?
Explain.

Exercise 5

The function $\displaystyle {f(x)}={x^3-3x^2}$ is defined in the interval $\displaystyle -1\leq x\leq4$ .

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

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

\displaystyle f(-1)

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

/ – 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

left extremum

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

/ – 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

right extremum

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

/ – 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(4)

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

/ – 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 $\displaystyle f$ in the interval, at the ends too, and the kind of each.
What is the absolute maximum, and what is the absolute minimum?

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
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 properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

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

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
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 instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

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
$\displaystyle y-y_1=m(x-x_1)$

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
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$