# Calculus

posted by .

Find a cubic function f(x) = ax^3 + bx^2 + cx + d that has a local maximum value of 3 at x = −2 and a local minimum value of 0 at x = 1.

• Calculus -

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.

• Calculus -

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

## Similar Questions

The function f(x)=4x+9x^-1 has one local minimum and one local maximum. Question: This function has a local minimum at x=______ with a value _______ and a local maximum at x=________ with value _______?
2. ### Calculus

I'm desperate.. I keep getting b=0 and I know it's wrong!! Find a cubic function f(x) = ax^3 + bx^2 + cx + d that has a local maximum at (-3,3) and a local minimum at (2,0) knowing that c=-12b and d=3-27b.
3. ### Calc

Find a cubic function, in the form below, that has a local maximum value of 3 at -2 and a local minimum value of 0 at 1. f (x) = ax3 + bx2 + cx + d
4. ### CALCULUS HELP ASAP!:(

Find a cubic function, in the form below, that has a local maximum value of 3 at -4 and a local minimum value of 0 at 3.
5. ### Calculus (help steve)

f(x)=−8x^3+6ax^2−3bx+4 has a local minimum at x=1 and a local maximum at x=3. If a and b are the local minimum and maximum, respectively, what is the absolute value of a+b?
6. ### Calculus

The function f(x)=-2x^3+30x^2-96x+8 has one local minimum & one local maximum. This function has a local minimum at x equals ______ with value __________ and a local maximum at x equals _______ with value __________ .
7. ### calculus 1

find the cubic function,f(x)=ax^3+bx^2+cx^d , has local maximum value of 3, at x=-2 and local minimum value of 0, at x=1.
8. ### Calculus

Find the local maximum and local minimum of the function. f(x) = x^3 − 12x^2 − 27x + 7
9. ### Calculus

Find a cubic function f(x)=ax^3+cx^2+d that has a local maximum value of 9 at -4 and a local minimum value of 6 at 0. Find a, c, and d.
10. ### Calculus

Find a cubic function f(x)=ax^3+cx^2+d that has a local maximum value of 9 at -4 and a local minimum value of 6 at 0. Find a, c, and d. I know that d = 6 but I am worked for hours trying to find a and c. Please help!!

More Similar Questions