Final answer:
The only option that is a representation of the differentiable function f, where f'(2) = 4, is B) f(x) = 4x + 5.
Step-by-step explanation:
The differential of a function f(x) gives us the slope of the tangent line to the graph of the function at a particular point. In this case, f'(2) = 4, meaning that the slope of the tangent line at x = 2 is 4.
Now let's look at each option:
A) f(x) = 2x² - 3x + 4
B) f(x) = 4x + 5
C) f(x) = eˣ
D) f(x) = ln(x)
A) f(x) = 2x² - 3x + 4: This is a quadratic function, and its derivative is f'(x) = 4x - 3. Evaluating it at x = 2 gives us f'(2) = 4(2) - 3 = 5, not 4, so A) is not a representation of the given differential.
B) f(x) = 4x + 5: The slope of a linear function is constant, so the derivative of this function is f'(x) = 4. Evaluating it at x = 2 gives us f'(2) = 4, which matches the given information. Therefore, B) is a representation of the given differential.
C) f(x) = eˣ: The derivative of the exponential function e^x is itself, so f'(x) = e^x. Evaluating it at x = 2 gives us f'(2) = e^2, not 4. Therefore, C) is not a representation of the given differential.
D) f(x) = ln(x): The derivative of the natural logarithm function ln(x) is 1/x, so f'(x) = 1/x. Evaluating it at x = 2 gives us f'(2) = 1/2, not 4. Therefore, D) is not a representation of the given differential.
In summary, the only option that is a representation of the differentiable function f, where f'(2) = 4, is B) f(x) = 4x + 5.