Question

Find f such that the given conditions are satisfied. f'(x) = x2 +3, f(3) = 42 ) O A. f(x)= x + 3x + 6 B. F(x) = 3 X3 + 3x O c. f(x)=x + 3x 3 + 3x + 24 X3 D. f(x) = + 3x + 24 3

Answer 1

D. f(x) = x^3 + 3x + 24

Explanation:

To find the function f(x) that satisfies the given conditions, we need to integrate f'(x) = x^2 + 3 to obtain f(x), and then substitute the value of f(3) = 42 to determine the constant of integration. Let's go through the steps:

Integration of f'(x):

∫(x^2 + 3) dx = (1/3)x^3 + 3x + C

Now, we substitute f(3) = 42 to determine the constant of integration:

(1/3)(3)^3 + 3(3) + C = 42

(1/3)(27) + 9 + C = 42

9 + 9 + C = 42

18 + C = 42

C = 42 - 18

C = 24

So, the function f(x) that satisfies the given conditions is:

f(x) = (1/3)x^3 + 3x + 24

Among the provided options, the correct answer is:

D. f(x) = x^3 + 3x + 24