Find f(x), if f “(x) = 1 – 6x, f ’(1) = 2 and f(-1) = 8.

Question

Find f(x), if f “(x) = 1 – 6x, f ’(1) = 2 and f(-1) = 8.

Answer 1

f'(x) = ∫ f"(x)dx = x - 3x² + C

Since f'(1) = 2, we have -2 + C = 2. So, C = 4.

Therefore, f'(x) = x - 3x² + 4

Integrate f'(x) to get f(x):

f(x) = (1/2)x² - x³ + 4x + K

Since f(-1) = 8, we get 1/2 + 1 - 4 + K = 8

So, K = 10.5

f(x) = (1/2)x² - x³ + 4x + 10.5