If:
0 = f(1) = a+b+c+d
2 = f(-2) = -8a+4b-2c+d
0 = f'(1) = 3a+2b+c
0 = f'(-2) = 12a-4b+c
"since we now have four equations and four unknowns, we can determine unique solution. Solving for "a" and "b", a=, b="
Please show the steps, I am so confused and no one knows how to do it.
Thank you so much in advance!
apparently f(x) = ax^3 + bx^2 + cx + d
so f'(x) = 3ax^2 + 2bx + c
and you were given f(1), f(-2) and f'(1), f'(-2)
take the last two equations, equate c:
3a+2b = 12a-4b
6b = 9a
2b = 3a
take the first two equations, equate d:
a+b+c = -8a+4b-2c
so,
2a+2b+2c = -16a+8b-4c
but 2b = 3a, so
2a+3a+c = -16a + 12a - 4c
9a = -5c
start substituting those values back in, and you get
a,b,c,d = 4/27, 2/9, -8/9, 14/27
y = 1/27 (4x^3 + 6x^2 - 24x + 14)
= 2/27(x-1)^2 (2x+7)
f(1) = 0 and f(-2) = 2
y' = 4/9 (x^2 + x - 2) = 4/9 (x+2)(x-1)
it's easy to see that f'(1) = f'(-2) = 0
To solve for the values of "a" and "b" in the given system of equations, we can use the method of substitution. Here are the steps:
Step 1: Start with the given equations:
0 = a + b + c + d (Equation 1)
2 = -8a + 4b - 2c + d (Equation 2)
0 = 3a + 2b + c (Equation 3)
0 = 12a - 4b + c (Equation 4)
Step 2: Solve Equation 3 for "c" in terms of "a" and "b":
c = -3a - 2b (Equation 5)
Step 3: Substitute Equation 5 into Equation 1 and Equation 2:
0 = a + b - 3a - 2b + d (Substituting Equation 5 into Equation 1)
= -2a - b + d (Equation 6)
2 = -8a + 4b - 2(-3a - 2b) + d (Substituting Equation 5 into Equation 2)
= -8a + 4b + 6a + 4b + d (Expanding)
= -2a + 8b + 6a + d (Simplifying)
= 4a + 8b + d (Equation 7)
Step 4: Solve Equation 7 for "d" in terms of "a" and "b":
d = 2 - 4a - 8b (Equation 8)
Step 5: Substitute Equation 8 into Equation 6:
0 = -2a - b + 2 - 4a - 8b (Substituting Equation 8 into Equation 6)
= -6a - 9b + 2 (Simplifying)
Step 6: Solve Equation 5 for "c" in terms of "a" and "b":
c = -6a - 9b + 2 (Equation 9)
Step 7: Now, we have three equations (Equation 5, Equation 8, and Equation 9) and three unknowns ("a", "b", and "c"). We can solve this system of equations using any method of solving systems of equations (elimination, substitution, or matrices).
By solving the system of equations, you will be able to find the values of "a", "b", and "c".
To solve for the unknowns "a" and "b" in the given system of equations, we will use the method of substitution. Here's a step-by-step explanation of how to solve it:
Step 1: Rewrite the given equations:
0 = f(1) = a + b + c + d ---- (Eq. 1)
2 = f(-2) = -8a + 4b - 2c + d ---- (Eq. 2)
0 = f'(1) = 3a + 2b + c ---- (Eq. 3)
0 = f'(-2) = 12a - 4b + c ---- (Eq. 4)
Step 2: Solve Eq. 3 for "c":
From Eq. 3, we have:
c = -3a - 2b ---- (Eq. 5)
Step 3: Substitute the value of "c" from Eq. 5 into Eq. 4:
0 = 12a - 4b + (-3a - 2b)
0 = 9a - 6b ---- (Eq. 6)
Step 4: Solve Eq. 1 for "d":
From Eq. 1, we have:
d = -a - b - c ---- (Eq. 7)
Substitute the value of "c" from Eq. 5 into Eq. 7:
d = -a - b - (-3a - 2b)
d = 2a + b ---- (Eq. 8)
Step 5: Substitute the values of "c" and "d" from Eq. 5 and Eq. 8 into Eq. 2:
2 = -8a + 4b - 2(-3a - 2b) + (2a + b)
2 = -8a + 4b + 6a + 4b + 2a + b
2 = 0a + 9b + 2a
2 = 11b ---- (Eq. 9)
Step 6: Solve Eq. 9 for "b":
From Eq. 9, we have:
b = 2/11
Step 7: Substitute the value of "b" into Eq. 6 to solve for "a":
0 = 9a - 6(2/11)
0 = 9a - 12/11
9a = 12/11
a = 12/99 = 4/33
Therefore, the solution is:
a = 4/33
b = 2/11
This is how you can solve the given system of equations to find the values of "a" and "b". Feel free to ask if you have any further questions!