Final answer:
To find the function f(x), first integrate f'(x) term by term. Using the given tangent line's y-intercept at x=0, determine the constant of integration as 4. The correct function is thus f(x) = sin(x) + e²⁸ + x² + 4 (Option A).
Step-by-step explanation:
To find a function f(x) given that f'(x) = cos(x) + e⁴²⁸ + 2x, and a tangent line at x = 0 given by y = 2x + 4, we must integrate f'(x) and then determine the constant of integration such that the tangent line criteria is met.
Integrating term by term, we get sin(x) + e⁴²⁸ + x² as the indefinite integral of f'(x). Since the derivative of a constant is 0, the most general form of f(x) will be sin(x) + e⁴²⁸ + x² + C, where C is the constant of integration that we need to determine.
The condition that the tangent line at x = 0 is y = 2x + 4 tells us two things about f(x) at x = 0: the value of the function f(0) and the slope of its tangent line, f'(0).
The slope of the tangent line at x = 0 is f'(0) = 2, which we know is true from our equation for f'(x). However, this is just to check the slope. What we are interested in is the value of the function at x = 0, which should be the same as the y-intercept of the tangent line, which is 4. This is because the equation of the tangent line is y = mx + b where m is the slope and b is the y-intercept. Plugging in x = 0 into the general form of f(x), we get f(0) = sin(0) + e⁴²·⁰ + 0² + C = C. For the tangent line to have a y-intercept of 4, we need C = 4.
Therefore, the correct function is f(x) = sin(x) + e⁴²⁸ + x² + 4.
So, we can rule out options B and D because a function with cos(x) would not integrate to give us the necessary terms for f'(x). We rule out option C because it does not contain the x² term that results from integrating 2x. Therefore, the correct answer is Option A: f(x) = sin(x) + e⁴²⁸ + x² + 4.