Posted by Benjamin on Monday, November 12, 2012 at 6:05pm.
At the local min/max points, the derivative is zero.
f' = 3ax^2 + 2bx + c = 0
Plug in x = -3 and x = 2, and that gives you two equations.
Also, use f = ax^3 + bx^2 + cx + d
Plug in the values (x = -3, f = 3) and (x = 2, f = 0).
That gives you two more equations.
Now solve the four simultaneous linear equations in a, b, c, and d.
f ' (x) = 3ax^2 + 2bx + c
f '(-2) = 0
12a -4b + c = 0 (#!1
f ' (1) = 0
3a + 2b + c = 0 (#2)
#1 - #2 ----> 9a - 6b = 0
or 3a = 2b
also (-2,3) lies on original
-8a + 4b - 2c + d = 3
and (1,0) lies on it
a + b + c + d = 0
subtract those two equations
9a -3b+3c = -3 ,or
3a - b + c = -1 , (#3)
#2 - #3 :
3b = 1
b = 1/3 , but a = 2b/3 = (2/3)(1/3) = 2/9
in #3
3(2/9) -1/3 + c = -1
c = -4/3
in a+b+c+d = 0
2/9 + 1/3 - 4/3 + d = 0
d = 7/9
so f(x) = (2/9)x^3 + (1/3)x^2 - (4/3)x + 7/9
Related Questions
Calculus - I'm desperate.. I keep getting b=0 and I know it's wrong!! ...
calculus - Find the cubic function f(x)=ax^3+bx^2+cx+d that has a local max ...
Calculus - A cubic polynomial function f is defined by f(x) = 4x^3 + ax^2 + bx...
Calculus - A cubic polynomial function f is defined by f(x) = 4x^3 + ax^2 + bx...
calculus - Find the cubic function of the form y=ax^3 + bx^2 + cx + d which has ...
Calc - Find a cubic function, in the form below, that has a local maximum value ...
POLYNOMIALS - The cubic polynomial f(x) is such that the coefficient of x^3 is -...
Calculus AB - Find a, b, c, and d such that the cubic function ax^3 + bx^2 + cx...
CALCULUS HELP ASAP!:( - Find a cubic function, in the form below, that has a ...
math - find the values of a and b if the function y=2x^3+ax^2+bx+36 has a local ...
For Further Reading
