MENU
❌
Home
Math Forum/Help
Problem Solver
Practice
Algebra
Geometry
Tests
College Math
History
Games
MAIN MENU
1 Grade
Adding and subtracting up to 10
Comparing numbers up to 10
Adding and subtracting up to 20
Addition and Subtraction within 20
2 Grade
Adding and Subtracting up to 100
Addition and Subtraction within 20
3 Grade
Addition and Subtraction within 1000
Multiplication up to 5
Multiplication Table
Dividing
Rounding
Perimeter
4 Grade
Adding and Subtracting
Equivalent Fractions
Divisibility by 2, 3, 4, 5, 9
Area of Squares and Rectangles
Fractions
Equivalent Fractions
Least Common Multiple
Adding and Subtracting
Fraction Multiplication and Division
Operations
Mixed Numbers
Decimals
Expressions
6 Grade
Percents
Signed Numbers
The Coordinate Plane
Equations
Expressions
Polynomials
Polynomial Vocabulary
Symplifying Expressions
Polynomial Expressions
Factoring
7 Grade
Angles
Linear Functions
8 Grade
Linear Functions
Systems of equations
Slope
Parametric Linear Equations
Word Problems
Exponents
Roots
Quadratic Equations
Vieta's Formulas
Progressions
Arithmetic Progressions
Geometric Progression
Progressions
Number Sequences
Reciprocal Equations
Logarithms
Logarithmic Expressions
Logarithmic Equations
Extremal value problems
Trigonometry
Geometry
Intercept Theorem
Slope
Law of Sines
Law of Cosines
Vectors
Analytic Geometry
Numbers Classification
Probability
Limits of Functions
Properties of Triangles
Pythagorean Theorem
Matrices
Complex Numbers
Inverse Trigonometric Functions
Home
Practice
Extremal value problems
Easy
Normal
Difficult
Extremal value problems: Problems with Solutions
Problem 1
Find the minimal value of the function [tex]f(x)=3x^3-9x^2+6[/tex] for [tex]x \in [-1;5][/tex].
Solution:
The minimal value could be in both ends of the interval or in a point, in which there is a local minimum for the function. We calculate [tex]f(-1)=3.(-1)-9+6=-6[/tex] and [tex]f(5)=3.5^3-9.5^2+6=5^2(3.5-9)+6=25.6+6>-6[/tex], which is obviously not the minimal value.
All that is left is to find the local extrema of the function. [tex]f'(x)=9x^2-18x=9x(x-2)[/tex], so there are extrema in [tex]x=0[/tex] and [tex]x=2[/tex]. [tex]f(0)=3.0-9.0+6=6[/tex], so it is not a minimal value. [tex]f(2)=3.8-9.4+6=-6[/tex]. So the minimal value is [tex]-6[/tex] and is reached in two points - [tex]x=-1[/tex] and [tex]x=2[/tex].
Problem 2
Find the maximum value of the function [tex]f(x)=x-5[/tex] if
x
is a number between
-5
and
13
.
Solution:
Since
f(x)
is a linear function whose slope is
1
, a positive number, it is strictly increasing for all
x
. Therefore its maximal value is reached for the largest value of
x
,
x=13
and
f(x)=13-5=8
.
Problem 3
Find the maximal value for the function [tex]f(x)=4sin(x)[/tex]
Solution:
It is known that [tex]-1 \le sin(x) \le 1[/tex]. We multiply this by 4 and get [tex]-4 \le 4sin(x) \le 4[/tex], so the maximal value for this function is 4.
Problem 4
Find the minimal value for the function [tex]f(x)=|x|-1[/tex].
Solution:
[tex]|x| \ge 0[/tex], we add
-1
to both sides of the inequality and get [tex]|x|-1 \ge -1[/tex].
Easy
Normal
Difficult
Submit a problem on this page.
Problem text:
Solution:
Answer:
Your name(if you would like to be published):
E-mail(you will be notified when the problem is published)
Notes
: use [tex][/tex] (as in the forum if you would like to use latex).
Correct:
Wrong:
Unsolved problems:
Contact email:
Follow us on
Twitter
Facebook
Author
Copyright © 2005 - 2019.