If f'(x) is decreasing, then f”(x) is?
If the derivative, f'(x), is decreasing, it means that the rate at which the function is changing is decreasing
If the derivative, f'(x), is decreasing, it means that the rate at which the function is changing is decreasing. In other words, the slope of the function is getting smaller as x increases.
Mathematically, if f'(x) is decreasing, it implies that f”(x), the derivative of f'(x), is negative. This is because the derivative of a decreasing function is negative, as it represents the rate of change of the derivative itself.
To understand this concept visually, consider a graph of a function f(x). If f'(x) is decreasing, it means that the slope of the tangent line to the graph is getting smaller and smaller as x increases. This indicates that the graph is curving downwards and concave down, which suggests a negative second derivative, f”(x).
In summary, if f'(x) is decreasing, then f”(x) will be negative, indicating that the function is concave down.
More Answers:
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The Relationship Between Increasing f'(x) and Positive f”(x)