Question

Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.) f ''(x) = 12x + 36x^2

Answer 1

To find the function f given the second derivative f''(x) = 12x + 36x^2, integrate the equation twice to get f(x) = 2x^3 + 3x^4 + Cx + D.

Step-by-step explanation:

To find the function f given the second derivative f''(x) = 12x + 36x^2, we need to integrate the given equation twice. Let's start with the first integration:

∫ f''(x) dx = ∫ (12x + 36x^2) dx

By integrating, we get:

f'(x) = 6x^2 + 12x^3 + C

Next, we integrate again:

∫ f'(x) dx = ∫ (6x^2 + 12x^3 + C) dx

After integration, we have:

f(x) = 2x^3 + 3x^4 + Cx + D

Therefore, the function f is given by f(x) = 2x^3 + 3x^4 + Cx + D, where C and D are constants.