Maximizing The Rate Of Change Of A Function: Critical Points And Inflection Points

the graph of f'(x) is differentiable and has a relative maximum at x=2

f(x) has a point of inflection at x=2

When f'(x) has a relative maximum at x=2, it means that the rate of change of f(x) is increasing up to x=2, and then decreasing afterwards. This also means f(x) is changing from increasing to decreasing at x=2, which indicates that x=2 is a critical point of f(x).

Since the graph of f'(x) is differentiable, it means that f(x) is continuous and its derivatives exist at x=2. Thus, the critical point at x=2 can be classified as a local maximum, since the slope of f(x) is positive before it and negative after it in a small interval around x=2.

To determine whether x=2 is also an absolute maximum, we need to examine the behavior of f(x) in a neighborhood of x=2. This can be done by analyzing the sign of the second derivative of f(x), denoted as f”(x).

If f”(x) > 0 for all x in a sufficiently small interval around x=2, then f(x) is concave-up, and x=2 is an absolute minimum. Conversely, if f”(x) < 0 for all x in a sufficiently small interval around x=2, then f(x) is concave-down, and x=2 is an absolute maximum. If the sign of f''(x) changes at x=2, then x=2 is an inflection point, where the curvature of the graph changes from convex to concave, or vice versa. Thus, to fully analyze the behavior of f(x) at x=2, we need to find f''(x) and examine its sign. This can be done by taking the second derivative of f(x), and evaluating it at x=2. Overall, the existence of a relative maximum of f'(x) at x=2 indicates a critical point of f(x), where the slope changes from increasing to decreasing. The nature of the critical point (local or absolute maximum/minimum, or an inflection point) depends on the behavior of f''(x) in a small interval around x=2.

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