Find f.
f ''(x) = −2 + 12x − 12x2, f(0) = 9, f '(0) = 14
d^2y/dx^2 = -12 x^2 + 12 x - 2
dy/dx = - 4 x^ 3 + 6 x^2 - 2 x + c
at = 0 dy/dx = 14 so c = 14
dy/dx = - 4 x^ 3 + 6 x^2 - 2 x + 14
y = - x^4 + 2 x^3 - x^2 + 14 x + d
at 0 , y = d = 9
so
y = - x^4 + 2 x^3 - x^2 + 14 x + 9
almost the same as
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keep integration to go from f''(x) to f'(x) to f(x)
make sure you find the value of the constant in each step
To find the function f(x), we need to integrate the given second derivative f''(x) twice.
Step 1: Integrate f''(x) to find f'(x)
∫[-2 + 12x - 12x^2] dx = -2x + 6x^2 - 4x^3 + C₁
where C₁ is the constant of integration.
Step 2: Integrate f'(x) to find f(x)
∫[-2x + 6x^2 - 4x^3 + C₁] dx = -x^2 + 2x^3 - x^4 + C₁x + C₂
where C₂ is the constant of integration.
Now we use the initial conditions provided to find the values of C₁ and C₂.
Given: f(0) = 9
Substituting x = 0 into the equation for f(x):
-(0)^2 + 2(0)^3 - (0)^4 + C₁(0) + C₂ = 9
C₂ = 9
Given: f'(0) = 14
Substituting x = 0 into the equation for f'(x):
-2(0) + 6(0)^2 - 4(0)^3 + C₁ = 14
C₁ = 14
Finally, we substitute the values of C₁ and C₂ back into the equation for f(x):
f(x) = -x^2 + 2x^3 - x^4 + 14x + 9
Therefore, the function f(x) is given by -x^2 + 2x^3 - x^4 + 14x + 9.
To find f(x), we will integrate the given second derivative of f(x) with respect to x twice. The process is as follows:
Step 1: Integrate f ''(x) once to find f '(x):
f '(x) = ∫[-2 + 12x - 12x^2] dx
= ∫(-2) dx + ∫(12x) dx - ∫(12x^2) dx
= -2x + 6x^2 - 4x^3 + C1 (where C1 is the constant of integration)
Step 2: Integrate f '(x) to find f(x):
f(x) = ∫[-2x + 6x^2 - 4x^3 + C1] dx
= ∫(-2x) dx + ∫(6x^2) dx - ∫(4x^3) dx + ∫C1 dx
= -x^2 + 2x^3 - x^4 + C1x + C2 (where C2 is the constant of integration)
Now, to determine the values of C1 and C2, we use the given initial conditions:
Given f(0) = 9:
9 = -(0)^2 + 2(0)^3 - (0)^4 + C1(0) + C2
9 = 0 + 0 - 0 + 0 + C2
Therefore, C2 = 9.
Given f '(0) = 14:
14 = -2(0) + 6(0)^2 - 4(0)^3 + C1
14 = 0 + 0 - 0 + C1
Therefore, C1 = 14.
Finally, substituting the values of C1 and C2 into f(x), we have:
f(x) = -x^2 + 2x^3 - x^4 + 14x + 9
Thus, the function f(x) is given by -x^2 + 2x^3 - x^4 + 14x + 9.