Question

Find f.
f''(x)=20 x³+12 x²+4, f(0)=8, f(1)=5

Answer 1

To find the function f(x), we integrate f''(x) twice to obtain f(x), then use the given initial conditions f(0)=8 and f(1)=5 to determine the constants of integration.

Step-by-step explanation:

The student is asking to find a function f(x) given its second derivative, f''(x), and initial conditions for f(x) at x=0 and x=1. The second derivative is given by f''(x) = 20x³ + 12x² + 4. To find f(x), we need to integrate the second derivative twice to get the original function.

Step-by-step explanation:

1. Integrate f''(x) once to get f'(x): Integrate 20x³ + 12x² + 4 to get f'(x) = ⅔x⁴ + 4x³ + 4x + C1, where C1 is the constant of integration.
2. Use the initial condition f(0)=8 to find f(x): Integrate f'(x) to get f(x). After integrating, we find f(x) = ⅔x⁵ + x⁴ + 2x² + C1x + C2.
3. Apply the given f(0)=8 to find C2: If you plug in 0 for x, the only constant remaining is C2, hence f(0)=C2=8.
4. Apply f(1)=5 to find C1: Substitute x=1 and f(1)=5 into the equation from step 2 to solve for C1.
5. Having found C1 and C2, write the final form of f(x) with these constants.