Find a Cubic Function in the following form: f(x) = ax^3 + bx^2 + cx + d?

Question

Find a cubic function, in the form below, that has a local maximum value of 3 at -3 and a local minimum value of 0 at 2.

f(x) = ax^3 + bx^2 + cx + d

I know how to solve these types of questions, but for this particular version, I have tried so many different ways, but have been unable to get the right answer.

On web assign, I can only try the question five times before I get a zero for that question and I have redone the question four times getting it wrong.

Please help!
Thank you!

Answer 1

                                     

Answer 2

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f(x) = ax³ + bx² + cx + 2

a(-3)³ + b(-3)² + c(-3) + d = 3 and a(2)³ + b(2)² + c(2) + d = 0

-27a + 9b - 3c + d = 3 and 8a + 4b + 2c + d = 0



f’(x) = 3ax² + 2bx + c

3a(-3)² + 2b(-3) + c = 0 and 3a(2)² + 2b(2) + c = 0

27a - 6b + c = 0 and 12a + 4b + c = 0



System of four linear equations with four unknowns:

-27a + 9b - 3c + d = 3

8a + 4b + 2c + d = 0

27a - 6b + c = 0

12a + 4b + c = 0

—————————> a = 6/125, b = 9/125, c = -108/125, d = 132/125



f(x) = 1/125 (6x³ + 9x² - 108x + 132 )

━━━━━━━━━━━━━━━━



or

f(x) = 6x³/125 + 9x²/125 - 108x/125 + 132/125



or

f(x) = 0.048x³+ 0.072x² - 0.864x + 1.056

Answer 3

f'(x) = 3ax^2+2bx+c

f'(-3) = 27a-6b+c = 0,

f'(2) = 12a+4b+c = 0,

f(-3) = -27a+9b-3c+d = 3,

f(2) = 8a+4b+2c+d = 0

a = 6/125, b = 9/125, c = -108/125, d = 132/125

f(x) = 6x^3/125+9x^2/125-108x/125+132/125

See Graph

https://www.desmos.com/calculator/wsje3stqsu

Answer 4

f(-3) = 3

==> -27a + 9b -3c + d = 3

f(2) = 0

==> 8a + 4b + 2c + d = 0

f'(x) = 3ax^2 + 2bx + c

==>f'(-3) = 0 = 27a - 6b +c

and

==> f'(2) = 0 = 12a + 4b +c

4 equations with 4 variables

solve for a, b, c, d

Answer 5

f(x) = ax^3 + bx^2 + cx + d--->f'(x)=3ax^2+2bx+c



local maximum value of 3 at -3 --->f(-3)=3 and f'(-3)=0 , set up 2 equations

local minimum value of 0 at 2 --->f(2)=0 and f'(2)=0



solve 4 equations for a,b,c,d