Understanding Mathematical Concepts: The Relationship Between Increasing First Derivative and Positive Second Derivative

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

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

If f'(x) is increasing, it means that the derivative of function f(x) is getting larger as x increases. In other words, the slope of the tangent line to the graph of f(x) is increasing as we move along the x-axis.

Now, let’s consider the second derivative of f(x), denoted as f”(x). The second derivative measures the rate at which the first derivative is changing.

If the first derivative, f'(x), is increasing, it implies that the slope of the tangent line is becoming steeper and steeper as x increases. This indicates that the graph of f(x) is curving upwards at an accelerated rate.

Mathematically, we can say that if f'(x) is increasing, then f”(x) is positive (greater than zero), because positive values of the second derivative indicate a concave up behavior.

To summarize:
– If f'(x) is increasing, it implies that f(x) is curving upwards.
– If f”(x) is positive, it denotes that the curve is concave up.
– Therefore, if f'(x) is increasing, then f”(x) is positive.

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