1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
emmasim [6.3K]
3 years ago
14

What is absolute extrema of cube root of x on I=[-3,8]

Mathematics
1 answer:
hichkok12 [17]3 years ago
3 0
<span>These are points where f ' = 0. Use the quiotent rule to find f '. 

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

Set f ' (x) = 0 and solve for x. 

f ' (x) = 0 = (2-2x^3) / (x^3+2)^2 

Multiply both sides by (x^3+2)^2 

(x^3+2)^2 * 0 = (x^3+2)^2 * [(2-2x^3)/(x^3+2)^2] 
0 = 2 - 2x^3 

Add 2x^3 to both sides 

2x^3 + 0 = 2x^3 + 2 - 2x^3 
2x^3 = 2 

Divide both sides by 2 

2x^3 / 2 = 2 / 2 
x^3 = 1 

Take cube roots of both sides 

cube root (x^3) = cube root (1) 
x = 1. This is our critical point 

2) Points where f ' does not exist. 

We know f ' (x) = (2-2x^3) / (x^3+2)^2 

You cannot divide by 0 ever so f ' does not exist where the denominator equals 0 

(x^3 + 2)^2 = 0. Take square roots of both sides 
sqrt((x^3+2)^2) = sqrt(0) 
x^3 + 2 = 0. Add -2 to both sides. 
-2 + x^3 + 2 = -2 + 0 
x^3 = -2. Take cube roots of both sides. 
cube root (x^3) = cube root (-2) 
x = cube root (-2). This is where f ' doesnt exist. However, it is not in our interval [0,2]. Thus, we can ignore this point. 

3) End points of the domain. 

The domain was clearly stated as [0, 2]. The end points are 0 and 2. 

Therefore, our only options are: 0, 1, 2. 

Check the intervals 

[0, 1] and [1, 2]. Pick an x value in each interval and determine its sign. 

In [0, 1]. Check 1/2. f ' (1/2) = (7/4) / (17/8)^2 which is positive. 

In [1, 2]. Check 3/2. f ' (3/2) = (-19/4) / (43/8)^2 which is negative. 

Therefore, f is increasing on [0, 1] and decreasing on [1, 2] and 1 is a local maximum. 

f (0) = 0 
f (1) = 1/3 
f (2) = 1/5 

Therefore, 0 is a local and absoulte minimum. 1 is a local and absolute
maximum. Finally, 2 is a local minimum. </span><span>Thunderclan89</span>
You might be interested in
Simplify <br><br> ab + ab - bc - ab + bc
Alex777 [14]
Ab^2 (-bc & +bc would cancel out) but then -ab so the answer would just be ab
5 0
3 years ago
the perimeter of an isosceles triangle is 45. find the length of the third side if each of the equal sides is 14cm long
77julia77 [94]

Answer:

17cm

Step-by-step explanation:

The two equal sides are 14cm long so 45 - 14 - 14 = 17

8 0
3 years ago
What is the volume of this cube? *4.2in
Lapatulllka [165]
I think that it is  8.4 possibly
3 0
3 years ago
A fire truck is responding to an emergency. It accelerates from 0 to 66 km/hr in 11 seconds. What is its rate of acceleration?
Ghella [55]

Answer:n 6

Step-by-step explanation:

Divide 66 by 11 and you get 6

8 0
3 years ago
Read 2 more answers
Help help help help pls​
Ronch [10]

Answer:

[24, 375]

Step-by-step explanation:

f(2) = 24

f(5) = 375

f(x) is increasing in the given domain so the range is simply given by the extremes.

6 0
3 years ago
Other questions:
  • At a party there were 40 guests. One-fifth were men. What fraction of guests were women?
    14·2 answers
  • The average of 2 numbers is 1/2. if one of the numbers is 11/17 then the other number is ___
    14·1 answer
  • What is the constant of proportionality between y and x in the graph?
    9·1 answer
  • Round 2.873 to the nearest tenth
    7·1 answer
  • Can I get get some help please!
    9·1 answer
  • Determine the intercepts of the line. <br> y-intercept: (___, ___)<br> x-intercept: (___,___)
    5·1 answer
  • (04.02 MC)
    6·1 answer
  • In rugged terrain, Martin can hike 1/8 Mile in 1/2 hour. At this rate, how far can Martin hike in 1 hour?
    12·1 answer
  • A) 15<br><br> B) 5<br><br> C) 12<br><br> D) 3
    6·2 answers
  • sally decided she wanted homework help. no one gave it to her. if over 100 people were on the app to help and 1 person helped ho
    8·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!