Understanding the Relationship Between Increasing Derivative and Positive Second Derivative: The Key to Steeper Graph Slopes

If f'(x) is increasing, then f”(x) is?

If f'(x) is increasing, it means that as x increases, the derivative f'(x) is becoming larger

If f'(x) is increasing, it means that as x increases, the derivative f'(x) is becoming larger. In other words, the slope of the graph of f(x) is getting steeper as x increases.

The second derivative, f”(x), represents the rate of change of the derivative f'(x). In simpler terms, it tells us how the slope of the graph of f(x) is changing.

If f'(x) is increasing, it means that the slope of the graph of f(x) is becoming larger as x increases. This implies that the rate of change of the derivative, f”(x), must be positive.

Therefore, if f'(x) is increasing, then f”(x) must be positive. Alternatively, we can say that if f”(x) is positive, then f'(x) is increasing. These two statements are equivalent.

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