Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.) f ''(x) = 24x3 − 18x2 + 10x f(x) =

Question

Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)

f ''(x) = 24x³ − 18x² + 10x

f(x) =

Answer 1

Integrate f'''(x) to get f'(x) = 6x⁴ - 6x³ + 5x² + C, where C is a constant.

Integrate f'(x) to get f(x) = (6/5)x⁵ - (3/2)x⁴ + (5/3)x³ + Cx + D, where D is a constant.

To check that this works, take the derivative of f(x) to get f'(x) = 6x⁴ - 6x³ + 5x² + C, and then take the derivative of f'(x) to get f''(x) = 24x³ - 18x² + 10x, as desired.

So f(x) = (6/5)x⁵ - (3/2)x⁴ + (5/3)x³ + Cx + D