Answer :
Final answer:
To determine the value of the function f(x) at x = 7, we apply the fundamental theorem of calculus to the given information. Adding the result of the integral of f′(x) from 3 to 7 to f(3) results in f(7) being 34.
Explanation:
The question at hand involves determining the value of a function f(x) at a specific point, given information about its derivative and a definite integral. We are told that f(3) = 14 and that the integral of f′(x) from 3 to 7 is 20. To find the value of f(7), we use the fundamental theorem of calculus, which states that the integral of a function's derivative over an interval gives us the change in the function's values over that interval. Therefore, the increase in the function's value from x = 3 to x = 7 is 20. Since we know the function's value at x = 3, we simply add this increase to find f(7):
f(7) = f(3) + ⁷∫₃f′(x)dx
= 14 + 20
= 34
The correct answer is a) 34.
