If f'(x) is decreasing, then f”(x) is?
If f'(x) is decreasing, it means that the derivative of the function f(x) is getting smaller as x increases
If f'(x) is decreasing, it means that the derivative of the function f(x) is getting smaller as x increases. This indicates that the slope of the tangent line to the graph of f(x) is decreasing.
The second derivative, f”(x), represents the rate of change of the slope of the tangent line. If f'(x) is decreasing, then f”(x) can have one of the following possibilities:
1. f”(x) is negative: If the second derivative is negative, it means that the rate of change of the slope is decreasing. In other words, the graph of f(x) is concave down. This can be observed by visualizing a “smiling” or “frowning” curve.
2. f”(x) is zero: If the second derivative is zero, it means that the rate of change of the slope is zero. In other words, the graph of f(x) is neither concave up nor concave down. This can be visualized by a straight horizontal line or a flat curve.
3. f”(x) is positive: If the second derivative is positive, it means that the rate of change of the slope is increasing. In other words, the graph of f(x) is concave up. This can be visualized by an “U” shaped curve.
So, to summarize, if f'(x) is decreasing, the second derivative f”(x) can be negative, zero, or positive, indicating whether the graph of f(x) is concave down, flat, or concave up, respectively.
More Answers:
Understanding the Relationship Between a Concave Up Function and a Positive Second DerivativeUnderstanding Concave Down Functions and Their Second Derivatives: Implications for Rate of Change
Understanding the Relationship between Increasing Derivatives and Curvature: Explaining the Connection between f'(x) and f”(x)