Final answer:
The antiderivative of f(x) = (3/7)x - 3 given that f(1) = 3 is (3/14)x² - 3x + (81/14), including the constant determined using the specific value of the function. Option number a is correct.
Step-by-step explanation:
The question asks us to find the antiderivative of the function f(x) = (3/7)x - 3, given a specific function value, f(1) = 3. To find the antiderivative, we integrate the function term by term. The antiderivative of (3/7)x is (3/14)x² since the general rule is to increase the exponent by 1 and divide by the new exponent. The antiderivative of a constant -3 is simply -3x. Adding a constant C is important because the integration process includes the possibility of any constant value.
As we are given that f(1) = 3, we can use this information to solve for C. Substituting x = 1 into the antiderivative, we get (3/14) - 3 + C = 3. Solving for C gives us C = 3 + 3 - (3/14), which simplifies to C = 6 - (3/14) or C = (81/14). Therefore, the correct antiderivative with the constant is (3/14)x² - 3x + (81/14).