Quantcast

A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

burhan101

  • 2 years ago

Determine and classify all critical points of each function on the interval -4<x<4 f(x) = x⁴-4x³

  • This Question is Closed
  1. johnweldon1993
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Critical points are found via taking the first derivative of the function...and setting it = to 0 So what is \[\frac{ d }{ dx }x^4-4x^3\]

  2. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    0=4x²(x-3)

  3. johnweldon1993
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    One step ahead of me...so you will have 2 critical points.. being...?

  4. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    0 and 3 ?

  5. johnweldon1993
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    That would be correct...Now do you have to classify if they are a max or a min or an inflection point?

  6. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    How do i find that out?

  7. Loser66
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    plug them into the original function and calculate its value. f(0) =? and f(3)=? compare them to give out the answer

  8. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    (3, -27)

  9. Loser66
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok, between them, which one is bigger? the bigger is local max, the smaller is local min. right?

  10. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes

  11. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what do you mean which one :S

  12. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    like between (x,y)

  13. Loser66
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    for your problem, so far you have 4 critical points, -4, 0, 3, 4, just plug them into the original function and make conclusion. done.

  14. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    where did -4 and 4 come from ?

  15. oldrin.bataku
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    $$f(x) = x^4-4x^3\\f'(x)=4x^3-12x^2=4x^2(x-3)\\4x^2(x-3)=0\implies4x^2=0,x-3=0$$... so we conclude \(x=0\) and \(x=3\) are our critical points. Both lie in our interval so classify them.

  16. oldrin.bataku
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    @burhan101 they're our endpoints, which you'd want to test when finding *absolute* extrema. @Loser66 had a minor misunderstanding

  17. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @oldrin.bataku what do i do with these points now?

  18. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    plug them into f(x)?

  19. oldrin.bataku
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    @burhan101 no you want to check the second derivative:$$f''(x)=12x^2-24x=12x(x-2)\\f''(0)=0\\f''(3)=12(3)=36>0$$... so our derivative is increasing near \(x=3\) meaning \(x=3\) is a relative minimum; since our derivative is neither increasing nor decreasing near \(x=0\) we find it's neither! (hence the even multiplicity)

  20. burhan101
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    why are we plugging in 3 ?

  21. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.