When f ‘(x) is increasing, f(x) is
When f ‘(x) is increasing, it indicates that the rate of change of the function f(x) is increasing as x increases
When f ‘(x) is increasing, it indicates that the rate of change of the function f(x) is increasing as x increases. This means that the function is getting steeper and steeper as we move along the x-axis.
In terms of the function itself, when f ‘(x) is increasing, it implies that f”(x) (the derivative of f ‘(x)) is positive. This means that the slope of the tangent line to the graph of f(x) is increasing.
Graphically, if you were to plot the function f(x) on a Cartesian coordinate system, when f ‘(x) is increasing, the graph of f(x) would appear to be curving upwards at an increasing rate. In other words, the function would look like it is bending upwards more and more as you move to the right along the x-axis.
Intuitively, this means that the values of f(x) are increasing at an increasing rate as x increases. In other words, as x becomes larger, the function is growing faster and faster. The slope of the secant lines connecting two points on the graph would also be increasing.
Overall, when f ‘(x) is increasing, it indicates that f(x) is getting steeper and its values are increasing at an increasing rate as x increases.
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