LemonmathSelf-study sheet

Grade 12 · Lesson 1.2

Maximum and minimum values

Name: Date: Time spent: min

Goals

Understand the maximum and minimum value of a function on a set, and tell them apart from local maxima and minima.
Find the maximum and minimum on a closed interval $[a, b]$ by comparing a few values.
Find the maximum and minimum on an open interval with a table of variations.
Apply this to polynomial, rational, square-root and trigonometric functions and to real-life problems.

Remember

$M=\max\limits_D f$ when $f(x)\le M$ for every $x\in D$ and $M$ is reached at some point of $D$ . Likewise for $\min$ .
A function continuous on $[a, b]$ always has a maximum and a minimum value there.
On a closed interval: roots of $f'$ (and points where $f'$ is undefined) inside $(a, b)$ → compute $f$ there and at $a$ , $b$ → compare.
On an open interval: draw the table of variations; there may be no maximum or no minimum.
After a substitution, find the correct range of the new variable.

Exercises

Exercise 1

Find the maximum and minimum values of $y=x^3-3x^2$ on $[-1, 3]$ .

Exercise 2

Find the maximum and minimum values of $y=\dfrac{x-1}{x+2}$ on $[0, 3]$ .

Exercise 3

Find the maximum and minimum values of $y=x^4-2x^2+3$ on $[0, 2]$ .

Exercise 4

Find the minimum value of $y=x^2+\dfrac{16}{x}$ on $(0, +\infty)$ .

Exercise 5

Consider $f(x)=-x^3+3x^2+2$ on $[-1, 3]$ . Is each claim true or false?

a) The equation $f'(x)=0$ has two roots in $[-1, 3]$ .

b) The maximum value of $f$ on $[-1, 3]$ is $6$ .

c) $f$ reaches its maximum value only at $x=2$ .

d) The minimum value of $f$ on $[-1, 3]$ is $2$ .

Exercise 6

Find the maximum value of $y=x\sqrt{9-x^2}$ on $[0, 3]$ (give the result as a decimal).

Exercise 7

A workshop makes $x$ chairs a week ( $10\le x\le 100$ ) at a total cost of $C(x)=x^2+100x+2500$ (thousand đồng). The average cost per chair is $A(x)=\dfrac{C(x)}{x}$ . How many chairs a week should the workshop make to have the lowest average cost? What is that cost?

Exercise 8 Review

Find the maximum and minimum values of $y=\cos 2x+2\cos x$ on $[0, \pi]$ .

Exercise 9 Review

Find the maximum and minimum values of $y=-x^2+4x+1$ on $[0, 5]$ in two ways: with the vertex of the parabola (grade 10) and with the derivative.

Exercise 10 Challenge

Find the maximum and minimum values of $y=x+\sqrt2\cos x$ on $\left[0, \dfrac{\pi}{2}\right]$ .

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. The maximum value of

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

[-1, 3]

17

.B.

-15

.C.

-8

.D.

12

Question 2. The minimum value of

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

[0, 2]

-1

.B.

1

.C.

2

.D.

0

Question 3. The minimum value of

y=x+\dfrac{4}{x-1}

(1, +\infty)

4

.B.

3

.C.

5

.D.

-3

Question 4. A function

y=f(x)

is continuous on

[-2, 4]

and

f'(x)>0

for every

x\in(-2, 4)

. Which statement is correct?

\max\limits_{[-2, 4]} f(x)=f(-2)

.B.

\max\limits_{[-2, 4]} f(x)=f(4)

.C.

\min\limits_{[-2, 4]} f(x)=f(4)

.D.

f

has no maximum value on

[-2, 4]

Part II. True or false

Question 1. Consider

f(x)=x^4-8x^2+3

[-1, 3]

a) The equation $f'(x)=0$ has three roots in $[-1, 3]$ .	T / F
b) $f(2)=-13$ .	T / F
c) The maximum value of $f(x)$ on $[-1, 3]$ is $3$ .	T / F
d) The sum of the maximum and minimum values of $f(x)$ on $[-1, 3]$ is $-1$ .	T / F

Part III. Short answer

Question 1. A workshop produces

x

tonnes of goods a month (

0\le x\le 20

) and makes a profit of

P(x)=-x^3+18x^2+135x

(million đồng). What is the greatest monthly profit, in million đồng?

Question 2. Find the maximum value of

y=\sqrt{x}\,(6-x)

[0, 6]

. Give your answer to 1 decimal place.

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

Hints (only when stuck)

Exercise 1.

What you need: on a closed interval, compare the values at the roots of $y'$ in $(-1, 3)$ and at the two endpoints.

Steps:

Solve $y'=3x^2-6x=0$ .
Compute $y$ at the roots and at $-1$ , $3$ .

Exercise 2.

What you need: if a function is monotonic on a closed interval, its maximum and minimum are at the endpoints.

Steps:

Compute $y'$ and look at its sign.
Compute $y(0)$ and $y(3)$ .

Exercise 3.

What you need: keep only the roots of $y'$ in the open interval $(0, 2)$ .

Steps:

Solve $y'=4x^3-4x=0$ .
Reject the roots not in $(0, 2)$ , then compare the values.

Exercise 4.

What you need: on an open interval, draw the table of variations to read off the minimum.

Steps:

Compute $y'=2x-\dfrac{16}{x^2}=\dfrac{2(x^3-8)}{x^2}$ .
Find the positive root of $y'$ and the sign on each side.

Exercise 5.

What you need: compare the values at the roots of $f'$ and at the two endpoints; a maximum can be reached at several points.

Steps:

Solve $f'(x)=-3x^2+6x=0$ .
Compute $f(-1)$ , $f(0)$ , $f(2)$ , $f(3)$ .

Exercise 6.

What you need: $\left(\sqrt{u}\right)'=\dfrac{u'}{2\sqrt u}$ and comparing values on a closed interval.

Steps:

Compute $y'$ and write it as one fraction: the numerator is $9-2x^2$ .
Find the root in $(0, 3)$ and compare with $y(0)$ , $y(3)$ .

Exercise 7.

What you need: find the minimum of $A(x)=x+100+\dfrac{2500}{x}$ on $[10, 100]$ .

Steps:

Compute $A'(x)$ and solve $A'(x)=0$ .
Compare the values at the root and at $10$ , $100$ .

Exercise 8.

What you need: the double-angle formula $\cos 2x=2\cos^2x-1$ (grade 11); for $x\in[0, \pi]$ , $\cos x\in[-1, 1]$ .

Steps:

Put $t=\cos x$ and write $y=g(t)$ .
Find the maximum and minimum of $g$ on $[-1, 1]$ .

Exercise 9.

What you need: the parabola $y=ax^2+bx+c$ with $a<0$ has its highest point at the vertex $x=-\dfrac{b}{2a}$ (see the quadratic function).

Steps:

Method 1: find the vertex, check that it is in $[0, 5]$ , then compare with the two endpoints.
Method 2: solve $y'=0$ and compare as usual.

Exercise 10.

What you need: comparing values on a closed interval; to compare irrational numbers, use approximations.

Steps:

Solve $y'=1-\sqrt2\sin x=0$ on $\left(0, \dfrac{\pi}{2}\right)$ .
Compute $y$ at the root and at both endpoints, then compare using approximations.

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