Final answer:
To sketch a function a(x) that decreases on (-∞, 3) and increases on (3, ∞), consider a quadratic function such as y = (x - 3)^2, which has a local minimum at x = 3.
Step-by-step explanation:
To sketch a function a(x) that decreases on the interval (-∞, 3) and increases on the interval (3, ∞), one can imagine a function that has a low point at x = 3. This suggests that x = 3 is a point of local minimum. A simple example would be a quadratic function that opens upwards, such as y = (x - 3)2.
A possible sketch would show the graph of the function decreasing as it approaches the point (3, 0) from the left, reaching the minimum point at (3, 0), where the derivative of the function equals zero. Then, as x values become larger than 3, the function begins to increase, symbolizing a positive derivative for values of x greater than 3.
A more detailed example involves a graph where you would plot a smooth curve that gently slopes downwards to the point (3, 0) and then slopes upwards as it moves to the right of this point.