Understanding the Relationship Between Increasing First Derivative and Positive Second Derivative | Implications for the Curvature of a Function

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

If the first derivative f'(x) of a function f(x) is increasing, it means that the slope of the tangent line to the graph of f(x) is also increasing as x increases

If the first derivative f'(x) of a function f(x) is increasing, it means that the slope of the tangent line to the graph of f(x) is also increasing as x increases.

To determine the behavior of the second derivative f”(x), we need to consider the rate of change of the slope of the tangent line. In other words, we want to know if the slope of the tangent line is getting steeper or shallower as x increases.

If f'(x) is increasing, it means that the slope of the tangent line is also increasing. This suggests that the rate of change of the slope is positive, which implies that f”(x) is positive.

Therefore, if f'(x) is increasing, then f”(x) is positive. In other words, the second derivative of f(x) is positive. This indicates that the graph of f(x) is concave up, with its curvature increasing as x increases.

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