Final answer:
The original function comparing f(x) and g(x) contains an inconsistency. Revising the question, for x^3, a positive second derivative implies a positive first derivative. For the additional question, the function that matches the description given at x = 3 is y = x^2, satisfying the condition of having a positive slope that decreases as x increases.
Step-by-step explanation:
The original question about the function f(x) = g(x) = x^3 contains a typo. Assuming the intent was to compare properties of functions f(x) and g(x), let's make corrections. However, the reference information provided (75) is about another function f(x) at x = 3, which has a positive value and a positive slope that is decreasing. We must treat each of these separately.
For the question about f(x) = g(x) = x^3, the statement is inherently illogical because f(x) and g(x) cannot simultaneously equal each other and different expressions. However, if we examine the function x^3 and its derivatives, we can determine that:
- g'(x) > 0 tells us nothing about the second derivative of f(x), since they are a single function, and we would need to differentiate twice to ascertain information about f''(x).
- g''(x) > 0 indicates that the slope of g(x) is increasing, which for x^3 would always be true, hence f'(x) > 0.
Regarding the additional information question (75), option b. y = x² fits the description given; it has a positive value at x = 3, the slope is positive (since the derivative, 2x, is positive for x > 0), and the magnitude of the slope (2x) decreases as x gets smaller. So the correct answer would be option b).