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
erica [24]
3 years ago
10

Determine where the function is increasing and where it is decreasing. (Enter your answers using interval notation. If an answer

does not exist, enter DNE.)
f(t) = (t3 + 3t2)3
increasing    

decreasing    
Mathematics
1 answer:
tankabanditka [31]3 years ago
3 0
<span>The function can only change from increasing to decreasing, and visa-versa at those points where the slope of the function is 0. And the slope of the function is determined by the first derivative of the function. So let's calculate the first derivative. f(t) = (t^3 + 3t^2)^3 f'(t) = d/dt[ (t^3 + 3t^2)^3 ] f'(t) = 3(t^3 + 3t^2)^2 * d/dt[ t^3 + 3t^2 ] f'(t) = 3(d/dt[ t^3 ] + 3 * d/dt[ t^2 ])(t^3 + 3t^2)^2 f'(t) = 3(3t^2 + 3 * 2t)(t^3 + 3t^2)^2 f'(t) = 3(3t^2 + 6t)(t^3 + 3t^2)^2 Simplify f'(t) = 3(3t^2 + 6t)(t^3 + 3t^2)^2 f'(t) = 3 * 3t(t + 2)(t^3 + 3t^2)^2 f'(t) = 9t(t + 2)(t^2(t + 3))^2 f'(t) = 9t(t + 2)t^4(t + 3)^2 f'(t) = 9t^5(t + 2)(t + 3)^2 And looking at the function, it becomes obvious that the roots (or inflection points) are at t = 0, t = -2, and t = -3. Now the only places where f(t) can switch directions is at those 3 inflection points. And at exactly those inflection points the curve is neither increasing, nor decreasing. If the slope of the function is positive, then its value is increasing, and if the slope is negative, then the function is decreasing. So all we need to do is calculate the value of the first derivative for any value between each inflection point plus one value smaller than the smallest inflection point and another value higher than the highest inflection point. Range from [-infinity, -3) f'(-4) = 18432 Since the value is positive, the function is increasing from [-infinity, -3) Range from (-3, -2) f'(-2.5) = 30.51758 Since the value is positive, the function is increasing from (-3, -2) Range from (-2, 0) f(-1) = -36 Since the value is negative, the function is decreasing from (-2, 0) Range from (0, infinity) f(1) = 64 Since the value is positive, the function is increasing from (0, +infinity) To summarize: increasing from [-infinity, -3) increasing from (-3,-2) decreasing from (-2,0) increasing from (0,infinity]</span>
You might be interested in
How can you use an equation to make a prediction from a pattern?
Vanyuwa [196]
Upon formation of an equation from a given pattern, we know how the variables in the patter are related. Using the equation, we can find the value of one of the missing variables if the rest are known and also predict the values of the pattern at given conditions.
An example:
y = 2x + 5
if we are to predict the value of y at x = 3, we simply substitute 3 into x
y = 2(3) + 5
= 11
8 0
2 years ago
Read 2 more answers
−2x = 4y + 6 2(2y + 3) = 3x − 5 What is the solution (x, y) to the system of equations above
Wewaii [24]

Answer:

(1;-2)

Step-by-step explanation:

x=-2y-3

2(2y+3)=-6y-9-5

10y=-20

y=-2

x=4-3=1

6 0
3 years ago
Find the greatest possible error for the given measurement if it is to the nearest 10 miles.
Leni [432]
<span>The greatest possible error in a given measurement is half of term's measuring unit; and more complex math stuff the answer is  5 miles.</span>
5 0
3 years ago
How do you find range
Tamiku [17]
First order the data from least to greatest, then subtract the smallest value from the largest value in the set.
5 0
3 years ago
Read 2 more answers
guys im in pain and im tired n im over it pls just help me so my mom will stop screaming i dont wanna do schoola anymore
muminat

Answer:

b

Step-by-step explanation:

it shows you the price of all the 80 flowers so the number 80 is extra information.

5 0
2 years ago
Other questions:
  • A circular rug has a diameter of 6 feet. What is the area of the rug?
    8·1 answer
  • MOney mONey money if you see it claim it free points :) better say thank you on my pf or i will hunt you when i die
    9·2 answers
  • A parking garage charges $4.50 for each hour you park. You have $18 in your pocket.
    12·2 answers
  • Solve for x: x = 7 = 16<br> A. x=9<br> B. X=23<br> C. X= 112
    8·2 answers
  • PLEASEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
    14·2 answers
  • jayson used 0.5 cups of kernals to make 16 cups of popcorn how many cupes of popcore would he make if he used 0.75 cups of kerna
    7·2 answers
  • the area of a childrens rectangular playground is 680 square meters. the length is 14 meters longer than the width. determine th
    13·1 answer
  • A recipe uses both carrots and onions. The recipe uses 4 times more carrots than onions. If the recipe uses a total of 25 ounces
    6·1 answer
  • F(n) = n² – 3 g(n) = 4n - 1 Find f[g(1)]
    5·1 answer
  • Guys please help me
    14·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!