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

- anonymous

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

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- anonymous

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

- anonymous

have you found the derivative?

- anonymous

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

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## More answers

- mathslover

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

- 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.

- mathslover

Can you now solve for a, b ? @Lachlan1996

- anonymous

wait wait, hold on youve lost me here.

- mathslover

lol, take your time. Sorry :)

- anonymous

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

- anonymous

so i dont think you can simultaneously solve it

- anonymous

because you are simultaneously solving a derivative and a point

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

ah yes i see

- 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.

- anonymous

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

- anonymous

wait no i mean 6b not 9b

- anonymous

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

- anonymous

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

- anonymous

uhh okie doke so a=0.5b-0.5

- anonymous

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

- anonymous

I got a=7, b=-15

- anonymous

oh no made an error forgot to subtract 6b

- mathslover

OH MY FAULT , sorry :(

- anonymous

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

- anonymous

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

- mathslover

Depends but I do think, Yep! :)

- anonymous

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

- anonymous

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

- mathslover

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

- anonymous

mmm yes my exam is in 5 weeks.

- anonymous

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

- mathslover

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

- anonymous

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

- anonymous

You're welcome.

- mathslover

Welcome! :)

- anonymous

uhhhh small problem

- anonymous

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

- mathslover

Try to do it again, I think calculation mistake?

- anonymous

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

- anonymous

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

- mathslover

hmn, Take care of these small mistakes .

- anonymous

whew! you scared me ;)

- anonymous

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

- anonymous

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

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