How do I find f or the derivative?

Question

Find f.

f''(x) = −2 + 12x − 12x²,    f(0) = 2,    f'(0) = 12

Answer 1

To find the function, given the derivative, you must go backwards from the way you did a derivative from a function.

In this case, you are given f ''(x) so you must ask yourself, "What would the function be, that if I took its derivative, I would end up with -2 + 12x - 12x²?

The answer would be f '(x) = -2x + 6x² - 4x³ + C because when you take the derivative of f '(x) you would get:

f ''(x) = -2 + 12x -12x²

The "C" represents any constant term (any number) because the derivative of any number is zero. To determine what the value of "C" is, you need to use the other information given:

f '(0) = 12 therefore 12 = -2(0) + 6(0)² - 4(0)³ + C or C = 12. Now you have the complete f '(x) definition:

f '(x) = -2x + 6x² - 4x³ + 12 (or re-write in standard form as f '(x) = -4x³ + 6x² - 2x + 12)

To determine f(x), repeat the process. Ask, "What would the function be, that if I took its derivative, I would end up with -2x + 6x² - 4x³ + 12. The answer is f(x) = -x² + 2x³ - x⁴ + 12x + C because when you take the derivative of it, you get -2x + 6x² - 4x³ + 12.

Again, to calculate "C", use the other information given, f(0) = 2:

2 = -(0)² + 2(0)³ - (0)⁴ + 12(0) + C or C = 2 giving you f(x) = -x² + 2x³ - x⁴ + 12x + 2 or, in standard form, f(x) = - x⁴ + 2x³ - x² + 12x + 2

To double check, take the derivative of this twice to make sure you get −2 + 12x − 12x²