When f ‘(x) is increasing, f(x) is
concave up
When f ‘(x) is increasing, it implies that the slope of the tangent line at f(x) is increasing as x increases. This means that the rate at which f(x) is changing with respect to x is increasing.
Since f'(x) represents the instantaneous rate of change of f(x) with respect to x at any given point x, when f ‘(x) is increasing, it means that the derivative of f(x) is becoming larger and approaching a maximum value. This indicates that the function f(x) is getting steeper, or increasingly more curved.
Therefore, when f ‘(x) is increasing, it indicates that f(x) is increasing at an increasing rate. In other words, the function f(x) is accelerating. This can be interpreted as the function having a concave up shape, which means its second derivative is positive.
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