integrals question

Question

Find f

f "(x) = 20x³ + 12x² + 4,     f(0) = 1,    f(1) = 3

Answer 1

If f "(x) = 20x³ + 12x² + 4, then by taking the anti-derivative we can find that f'(x) = 5x⁴ + 4x³ + 4x + C₁ and we can again take the anti-derivative of f'(x) to find that f(x) = x⁵ + x⁴ + 2x² + C₁x + C₂

We can now use f(0) to say:

f(0) = 1 = (0)⁵ + (0)⁴ + 2(0)² + C₁(0) + C₂ to find that C₂ = 1

We can then plug that into the f(x) equation we have and use f(1) to calculate C₁ like this:

f(1) = 3 = (1)⁵ + (1)⁴ + 2(1)² + C₁(1) + 1

3 = 1 + 1 + 2 + C₁ + 1

3 = 5 + C₁

C₁ = -2

So f(x) = x⁵ + x⁴ + 2x² - 2x + 1