Extremum problems with polynomials

Calculus, grade 10 (5 units) Lesson 16 of 26Learning

Extremum problems with polynomials

The steps of a solution

An extremum problem asks when some quantity – an area, a volume, a profit, a height – is greatest or least. The derivative answers it once the quantity is written as a function:

Choose a variable $\displaystyle x$ , and write all the lengths or amounts with it.
Write the quantity to make greatest or least as a function of $\displaystyle x$ .
Fix its domain: which values of $\displaystyle x$ are possible in the problem.
Differentiate, solve $\displaystyle {f'(x)}={0}$ , and check by a table of signs that it is a maximum or a minimum.
Answer the question asked, in words and with units.

A fence along a wall

The example below finds the largest rectangular garden that 60 metres of fence can enclose, when one of its sides is a wall that needs no fence.

$\displaystyle x$ – the length of each of the two sides perpendicular to the wall, in metres. The side parallel to the wall: $\displaystyle 60-2x$ .
The area: $\displaystyle {A(x)}={x(60-2x)}={60x-2x^2}$ .
The domain: all the sides are positive, so $\displaystyle 0<x<30$ .
$\displaystyle {A'(x)}={60-4x}={0}$ where $\displaystyle {x}={15}$ . Before 15 the derivative is positive and after it negative – a maximum.
$\displaystyle {A(15)}={15\cdot30}={450}$ .

The largest garden: 15 metres by 30 metres, with an area of 450 m².

This problem can also be solved without the derivative: $\displaystyle {A(x)}={-2x^2+60x}$ is a parabola opening downwards, and its maximum is at the vertex, $\displaystyle {x}={\frac{-60}{2\cdot(-2)}}={15}$ .

An open box

From a square sheet of card with a side of 24 cm, a square with a side of $\displaystyle x$ is cut from each corner, and the edges are folded up into a box with no lid. The example below finds the greatest volume of the box.

The base is a square with a side of $\displaystyle 24-2x$ , and the height is $\displaystyle x$ : $\displaystyle {V(x)}={x(24-2x)^2}$ .
The domain: $\displaystyle 0<x<12$ .
By the product rule and the rule for a composite function: $\displaystyle {V'(x)}={(24-2x)^2+x\cdot2(24-2x)\cdot(-2)}={(24-2x)(24-6x)}$ .
$\displaystyle {V'(x)}={0}$ where $\displaystyle {x}={12}$ (outside the domain) or $\displaystyle {x}={4}$ . Before 4 the derivative is positive and after it negative – a maximum.
$\displaystyle {V(4)}={4\cdot16^2}={1024}$ .

The greatest volume is 1024 cm³, when squares with a side of 4 cm are cut out.

A closed domain and the meaning of the extremum

A ball is thrown upwards, and its height in metres after $\displaystyle t$ seconds is $\displaystyle {h(t)}={20t-5t^2}$ , until it lands after 4 seconds. The domain is closed: $\displaystyle 0\leq t\leq4$ .

$\displaystyle {h'(t)}={20-10t}={0}$ where $\displaystyle {t}={2}$ , and $\displaystyle {h(2)}={20}$ .
At the ends: $\displaystyle {h(0)}={0}$ and $\displaystyle {h(4)}={0}$ .
So the greatest height is 20 metres, after 2 seconds. At that moment the speed $\displaystyle h'(2)$ is 0: the ball stops rising and starts to fall.

In a closed domain the absolute extremum can be at an end too, so always compare the values at the ends with the values at the points where $\displaystyle {f'(x)}={0}$ .

Exercise 1

The sum of two positive numbers is 12.
$\displaystyle x$ – one of the numbers; the other: $\displaystyle 12-x$ .
$\displaystyle x$ is squared and multiplied by the other number: $\displaystyle {f(x)}={x^2(12-x)}$ .

Find $\displaystyle f'(x)$ .

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

/ – 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 $\displaystyle {f'(x)}={0}$ .

Work out $\displaystyle f(8)$ .

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

/ – 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

Which two numbers give the greatest product?
Explain why it is a maximum.

\displaystyle x

/ – 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 12-x

/ – 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

Exercise 2

A rectangle lies under the parabola $\displaystyle {y}={12-x^2}$ : two of its vertices are on the $\displaystyle x$ -axis, at the points $\displaystyle (-x,0)$ and $\displaystyle (x,0)$ , and the other two are on the parabola.
The area of the rectangle: $\displaystyle {A(x)}={2x(12-x^2)}$ .

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

Find $\displaystyle A'(x)$ .

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

/ – 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

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

Work out the greatest area.

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

/ – 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

What is the domain of $\displaystyle x$ ?
What are the sides of the largest rectangle?

\displaystyle 2x

/ – 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 12-x^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

Exercise 3

From a rectangular sheet with sides of 16 cm and 10 cm, a square with a side of $\displaystyle x$ is cut from each corner, and it is folded into a box with no lid.
The volume: $\displaystyle {V(x)}={x(16-2x)(10-2x)}$ .

Find $\displaystyle V'(x)$ .

$\displaystyle V(x)=x(16-2x)(10-2x)$

/ – 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 $\displaystyle {V'(x)}={0}$ .

What is the domain of $\displaystyle x$ ?
Which solution from part b is left?

\displaystyle x

/ – 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

Work out the greatest volume.

$\displaystyle V(x)=x(16-2x)(10-2x)$

/ – 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

Exercise 4

A factory makes $\displaystyle x$ hundred products a day, and $\displaystyle 0\leq x\leq8$ .
The daily profit, in thousands of shekels: $\displaystyle {P(x)}={-x^3+6x^2+15x}$ .
The derivative: $\displaystyle {P'(x)}={-3x^2+12x+15}$ .

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

Work out the profit at the point you found in the domain, and at the end $\displaystyle {x}={8}$ .

extremum

$\displaystyle P(x)=-x^{3}+6x^{2}+15x$

/ – 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 P(8)

$\displaystyle P(x)=-x^{3}+6x^{2}+15x$

/ – 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

How many products should the factory make a day, and what is the profit then?
What does $\displaystyle P(8)$ mean?

Exercise 5

A ball is thrown upwards.
Its height in metres after $\displaystyle t$ seconds: $\displaystyle {h(t)}={1.5+14t-5t^2}$ .

Find $\displaystyle h'(t)$ , the speed of the ball.

$\displaystyle h(t)=1.5+14t-5t^{2}$

/ – 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

Solve the equation $\displaystyle {h'(t)}={0}$ : when does the ball stop rising?

What is the greatest height of the ball?

$\displaystyle h(t)=1.5+14t-5t^{2}$

/ – 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

Exercise 6

Again a garden along a wall, with 60 metres of fence: $\displaystyle {A(x)}={x(60-2x)}$ .
This time the sides perpendicular to the wall can be at most 10 metres long, so $\displaystyle 0<x\leq10$ .

$\displaystyle {A'(x)}={60-4x}$ .
Solve the inequality $\displaystyle A'(x)>0$ .

Work out $\displaystyle A(10)$ .

$\displaystyle A(x)=x(60-2x)$

/ – 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

What is the greatest area now?
Why is it not at the point where $\displaystyle {A'(x)}={0}$ ?

/ – 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

Formulas: calculus, grade 10 (5 units)

Open as a page

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
$\displaystyle y=b$