## Lachlan1996 3 years ago given that f(x) = ax^3 + bx^2, f(2)=-4 and f'(3)=99 find f(x), f(3) and f'(2)

1. Lachlan1996

i cant find a way to isolate a or b to solve

2. tjones89

have you found the derivative?

3. Lachlan1996

the derivative is 3ax^2 + 2bx but that doesnt help solve it

4. mathslover

f(x) = ax^3 + bx^2 f(2) = 8a + 4b = -4 f(3) = 27a + 9b = 99

5. mathslover

8a + 4b = -4 27a+9b=99 from equation 1 , we have : a = $$\LARGE\frac{-4-4b}{8}=\frac{-1-b}{2}$$ put this value in the second equation.

6. mathslover

Can you now solve for a, b ? @Lachlan1996

7. Lachlan1996

wait wait, hold on youve lost me here.

8. mathslover

lol, take your time. Sorry :)

9. Lachlan1996

uhhh i got 27a + 6b =99 but that is the gradient as in the derivative

10. Lachlan1996

so i dont think you can simultaneously solve it

11. Lachlan1996

because you are simultaneously solving a derivative and a point

12. EulerGroupie

Yes! Now use the f(2)=-4 in the original equation to get your second equation with the two unknowns... a and b

13. EulerGroupie

Two equations with two unknowns should be solvable. Pick your favorite method for solving systems... @mathslover is using substitution.

14. Lachlan1996

but he is solving a f' (x) with a f (x)

15. EulerGroupie

a and b will be the same in the original and the derivative

16. Lachlan1996

ah yes i see

17. mathslover

f(2) = 8a + 4b = -4 f(3) = 27a + 9b = 99 a = (-1-b)/2 put this in f(3) equation. 27(-1-b)/2 + 9b = 99 (-27-27b+18b)=198 -27 -9b = 198 -9b = 198+27 = 225 b = -225/9 b = -25 You can solve for a now.

18. Lachlan1996

in the above equation isnt it meant to be 6a not 9a though?

19. Lachlan1996

wait no i mean 6b not 9b

20. EulerGroupie

Its actually a really cool way to come up with a system... one eq from the original and one from the derivative. Slick, huh?! correct... I got 6b rather than 9b as well

21. EulerGroupie

27a+6b=99 8a+ 4b=-4

22. Lachlan1996

uhh okie doke so a=0.5b-0.5

23. Lachlan1996

however substituting that into 27a + 6b = 99 i got 8.333333 as being b

24. EulerGroupie

I got a=7, b=-15

25. Lachlan1996

oh no made an error forgot to subtract 6b

26. mathslover

OH MY FAULT , sorry :(

27. Lachlan1996

no, no it was a simple mistake, i just wasnt sure if you had made a mistake or were doing something else

28. Lachlan1996

this seems like alot of work just to start off the question, do you think this would be in an exam?

29. mathslover

Depends but I do think, Yep! :)

30. EulerGroupie

For calculus, it doesn't seem out of bounds for a test.

31. Lachlan1996

ahh god, in a year 11 maths exam? i think im buggered then

32. mathslover

Whether it will come or not in exam but I think you must be PREPARED

33. Lachlan1996

mmm yes my exam is in 5 weeks.

34. EulerGroupie

Creating a small system and solving it with substitution is fairly standard in calculus.

35. mathslover

You have much time for it, go through these type of questions and ask your problems here...

36. Lachlan1996

okie doke, thanks very much for the help mate, its much appreciated. thankyou to you both

37. EulerGroupie

You're welcome.

38. mathslover

Welcome! :)

39. Lachlan1996

uhhhh small problem

40. Lachlan1996

we got the answer wrong? its meant to be y=7x^3 -15x^2

41. mathslover

Try to do it again, I think calculation mistake?

42. EulerGroupie

yes... a=7, b= -15

43. Lachlan1996

ah yes, last step of my working didnt put the -ve in the 15

44. mathslover

hmn, Take care of these small mistakes .

45. EulerGroupie

whew! you scared me ;)

46. Lachlan1996

will do, doesnt help when you get frustrated but. maths is not easy for me

47. sauravshakya

I think it is f'(3)=99 not f(3)=99