Final answer:
The function f(x) at x = 3 has a positive value and positive slope that decreases as x increases, which aligns with the properties of a quadratic function, not a linear one. Therefore, y = x² is the function that corresponds to the student's description. The correct option is (b) y = x².
Step-by-step explanation:
The student's question is regarding the interpretation of a twice-differentiable function f(x) and certain properties associated with its derivatives. In calculus, f'(x) represents the slope of the tangent line to the curve at a particular point, which describes the instantaneous rate of change of the function. The second derivative, denoted as f''(x), provides information about the concavity of the function and the rate at which the slope itself is changing. Although the function given may have a y-intercept equal to f(0), this does not inherently mean that the function is linear, as option C suggests. It's essential to examine the behavior of the function and its derivatives to determine its characteristics.
Considering the options provided to describe a function f(x) where, at x = 3, the value of f(x) is positive with a positive slope that is decreasing, we need to evaluate which option aligns with this behavior. Option (a) y = 13x describes a linear function with a constant slope of 13, which does not align with a decreasing slope as x increases. Option (b) y = x², however, is a quadratic function where the slope does decrease as x increases from the point where x is positive, fulfilling the stated conditions. As such, the correct option is (b) y = x².