Find 𝑓 ( 𝑥) if 𝑓′′(𝑥 ) = 7 − 9𝑥 + 8 𝑥^ 2, 𝑓 (0) = 9 and 𝑓′ (0) = −4.

Question

Find 𝑓 ( 𝑥) if 𝑓′′(𝑥 ) = 7 − 9𝑥 + 8 𝑥^ 2, 𝑓 (0) = 9 and 𝑓′ (0) = −4.

Answer 1

ƒ''(x) = 7 - 9x + 8x²

Integrating, we get

ƒ'(x) = ∫ (7 - 9x + 8x²) dx + C

ƒ'(x) = 7x - 9/2 x² + 8/3 x³ + C

Using the condition 𝑓′ (0) = −4, we have C = -4. Therefore

ƒ'(x) = 7x - 9/2 x² + 8/3 x³ - 4

Integrating again, we have

ƒ(x) = ∫ (7x - 9/2 x² + 8/3 x³ - 4) dx + D

ƒ(x) = 7/2 x² - 3/2 x³ + 2/3 x⁴ - 4x + D

Using the condition 𝑓(0) = 9, we have D = 9. Therefore

ƒ(x) = 7/2 x² - 3/2 x³ + 2/3 x⁴ - 4x + 9

or

ƒ(x) = 9 - 4x + 7/2 x² - 3/2 x³ + 2/3 x⁴