Let f(x )= ax^3 + bx^2 + cx + d. Determine the values of a, b, c and d if f(x) has critical points at x = -1 and x = 2, f(0) = 1 and f ′(0) = 6.

Question

Hello, please provide a detailed explanation per step to study. Thank-you!

Answer 1

Given:

f(x) = ax³ + bx² + cx + d

Critical points: x = -1, 2

f(0) = 1

f ^' (0) = 6

c and d can be found quickly. Simply use f(0) to solve for d and f ^' (0) to solve for c. (Hint: derive f(x))

Critical points are x-values which make a function's derivative either 0 or extend to infinity. We will assume that our critical points test for finite derivatives of f(x). Why? If f ^' (-1) and/or f ^' (2) extend to infinity, either both a and b are infinitely great, only a is infinitely great, or only b is infinitely great. Thus, there is ambiguity and testing for when f ^' (-1) and f ^' (2) are equal to 0 is infinitely more plausible.

f ^' (-1) = 0 AND f ^' (2) = 0

Using f ^' (x) = 3ax² + 2bx + c,

3a - 2b + 6 = 0

12a + 4b + 6 = 0

Setting both equations equal to each other and solving for b yields...

b = (-3/2)a

Substituting (-3/2)a into b of either equation (I will use the first) gives us...

3a - 2(-3/2)a + 6 = 0

3a + 3a + 6 = 0

6a + 6 = 0

6a = -6

a = -1

Using either equation to find b yields...

3(-1) - 2b + 6 = 0

6 - 3 - 2b = 0

3 - 2b = 0

2b = 3

b = 3/2

Thus, a, b, c, and d are found...

a = -1

b = 3/2

c = 6

d = 1

Our complete f(x) is...

f(x) = -x³ + (3/2)x² + 6x + 1

Answer 2

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

f'(0) = 6

3a(0)^2 + 2b*0 + c = 6

c = 6

f(0) = 1

a(0)^3 + b(0)^2 + c*0 + d = 1

d = 1

ax^3 + bx^2 + 6x + 1 = f(x)

f'(x) = 3ax^2 +2bx +6

The critical points are points at which f'(x) does not exist or it = 0

x=-1 is a critical point

f'(-1) = 0

f'(-1) = 3a(-1)^2 + 2b(-1) + 6 = 0

3a - 2b + 6 = 0

f'(2) = 0

f'(2) = 3a(2)^2 +2b(2) +6 = 0

12a + 4b + 6 = 0

Multiply the first equation by 2, and add it to the second equation.

6a - 4b + 12 = 0

18a + 18 = 0

18a = -18

a = -1

Plug that in to either equation.

12a + 4b + 6 = 0

12*-1 + 4b + 6 = 0

-12 + 4b + 6 = 0

4b - 6 = 0

4b = 6

b = 6/4 = 3/2