Question

Find f (x) if f'(x) = 3x² + 8x + 2 and f(1) = 2.

Answer 1

We can find f(x) by integrating f'(x) with respect to x. Using the power rule of integration, we get:

f(x) = x^3 + 4x^2 + 2x + C

where C is the constant of integration.

To find the value of C, we use the fact that f(1) = 2. Substituting x = 1 and f(x) = 2 into the equation above, we get:

2 = 1^3 + 4(1)^2 + 2(1) + C
2 = 7 + C
C = -5

Therefore, the function f(x) is:

f(x) = x^3 + 4x^2 + 2x - 5