The Relationship Between Increasing f'(x) and Positive or Zero f”(x)

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

If f'(x) is increasing, then f”(x) can either be positive or equal to zero

If f'(x) is increasing, then f”(x) can either be positive or equal to zero.

Let’s break down the statement:

1. If f'(x) is increasing, it means that as x increases, the derivative f'(x) is also increasing. This implies that the slope of the tangent line to the graph of f(x) is getting larger as x increases.

2. The second derivative, f”(x), measures the rate at which the derivative, f'(x), is changing as x varies. In other words, it describes whether the slope of the tangent line is increasing or decreasing.

Now, let’s consider the possibilities:

– If f”(x) is positive, it means that as x increases, the slope of the tangent line is becoming steeper (i.e., increasing). This is consistent with f'(x) being increasing, as stated in the question. So, it is possible for f”(x) to be positive when f'(x) is increasing.

– If f”(x) is zero, it means that the slope of the tangent line is not changing as x varies. In this case, f'(x) is constant, which is also consistent with f'(x) being increasing. So, it is also possible for f”(x) to be zero when f'(x) is increasing.

It is important to note that f”(x) cannot be negative when f'(x) is increasing. If f”(x) were negative, it would imply that the slope of the tangent line is decreasing (i.e., becoming less steep) as x increases. This would contradict the initial assumption that f'(x) is increasing.

In summary, if f'(x) is increasing, then f”(x) can be either positive or equal to zero, but it cannot be negative.

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