Final answer:
The function A(x), which is an integral of f(t), decreases when f(t) is negative over the interval being considered. If f(t) is a negative constant or a function like -x^2, with x > 0, A(x) will decrease as 'x' increases because the areas contributing to the integral are negative.
Step-by-step explanation:
The student's question concerns the behavior of the function A(x) represented by the integral of another function f(t). Specifically, when the function A is decreasing. Since A is the antiderivative of function f, A will be decreasing where f(t) is negative because the value of the integral will decrease as 'x' increases. In other words, if we consider A(x) = ∫ f(t) dt from a to x, the function A decreases when f(t) < 0 for t in the interval (a, x). The accumulation of negative values of f(t), which represent the area under the curve f(x) from x1 to x2, contributes to the decrement of A(x).
To better understand the concept, let's consider an example: If f(x) = -2, which is a negative constant function, then A(x) would be a linear function with a negative slope. This would result in A(x) decreasing as x increases. Similarly, if f(x) is a function like -x^2, which is always negative for x ≠ 0, then the integral A(x) will also be a decreasing function for x > 0.