If (f ‘ ‘(x) = 2), (f ‘ (1) = 0) and (f(0) = – 8), find f(x).
Explanation
(f ‘ ‘ (x) = frac{mathrm d ^{2} y}{mathrm d ^{2} x} = 2)
(f ‘ (x) = int 2 mathrm {d} x)
= (2x + c)
When x = 1, f'(x) = 0.
(2(1) + c = 0 implies c = -2)
(therefore f ‘ (x) = 2x – 2)
(f(x) = int (2x – 2) mathrm {d} x)
= (x^{2} – 2x + c)
When x = 0, f(x) = -8
(0^{2} – 2(0) + c = -8)
(c = -8)
(therefore f(x) = x^{2} – 2x – 8)