Question

Find f. (use c for the constant of the first antiderivative and d for the constant of the second antiderivative.) f ³(x) = 16x³-30x + 1 f(x)=________

Answer 1

To find f(x), we need to antidifferentiate the given function f³(x) = 16x³ - 30x + 1. Applying the power rule for integration, we find f(x) = (1/4)x⁴ - 15x² + x + c.

Step-by-step explanation:

To find f(x), we need to antidifferentiate the given function f³(x) = 16x³ - 30x + 1.

Antidifferentiation is the reverse process of differentiation. We will apply the power rule for integration and integrate each term of the function.

For 16x³, we apply the power rule which states that the integral of xⁿ is (1/(n+1))x^(n+1).

Here, n = 3, so we have (1/4)x⁴ as the antiderivative of 16x³.

For -30x, the antiderivative is (-15x²) by applying the power rule with n = 1.
The antiderivative of 1 is x + c, where c is the constant of integration.

Combining all the antiderivatives, we get f(x) = (1/4)x⁴ - 15x² + x + c.

This is the function f(x) that corresponds to the given f³(x).