When f ‘(x) is increasing, f(x) is
When f ‘(x) is increasing, it means that the derivative of the function f(x) is also increasing
When f ‘(x) is increasing, it means that the derivative of the function f(x) is also increasing. This implies that the rate at which f(x) is changing with respect to x is increasing.
Intuitively, this means that as x increases, the slope or steepness of the graph of f(x) is getting larger. Graphically, you can think of this as the tangent line to the graph of f(x) becoming steeper as x increases.
Mathematically, if f ‘(x) is increasing, it implies that the second derivative f”(x) is positive for the given range. The second derivative measures the rate of change of the derivative, which in this case is positive. This indicates that the slope of the function is continuously increasing, which translates to f(x) also becoming steeper as x increases.
In terms of the behavior of f(x), when f'(x) is increasing, it suggests that f(x) is accelerating. This could manifest as the graph of f(x) curving upwards more and more steeply as x increases, indicating a higher rate of change of the function.
Overall, when f ‘(x) is increasing, it implies that f(x) is getting steeper or faster, and its rate of change is increasing as x increases.
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