burhan101 Group Title Determine and classify all critical points of each function on the interval -4<x<4 f(x) = x⁴-4x³ one year ago one year ago

1. johnweldon1993 Group Title

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 Group Title

0=4x²(x-3)

3. johnweldon1993 Group Title

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

4. burhan101 Group Title

0 and 3 ?

5. johnweldon1993 Group Title

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

6. burhan101 Group Title

How do i find that out?

7. Loser66 Group Title

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

8. burhan101 Group Title

(3, -27)

9. Loser66 Group Title

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

10. burhan101 Group Title

yes

11. burhan101 Group Title

what do you mean which one :S

12. burhan101 Group Title

like between (x,y)

13. Loser66 Group Title

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 Group Title

where did -4 and 4 come from ?

15. oldrin.bataku Group Title

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

16. oldrin.bataku Group Title

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

17. burhan101 Group Title

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

18. burhan101 Group Title

plug them into f(x)?

19. oldrin.bataku Group Title

@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 Group Title

why are we plugging in 3 ?