If f'(x) is decreasing, then f”(x) is?
If f'(x) is decreasing, it means that the derivative of the function f(x) with respect to x, denoted as f'(x), is itself decreasing
If f'(x) is decreasing, it means that the derivative of the function f(x) with respect to x, denoted as f'(x), is itself decreasing. In other words, as x increases, the value of f'(x) is getting smaller.
Now, f”(x) represents the second derivative of the function f(x) with respect to x. The second derivative measures the rate at which the slope of the function is changing. To determine the behavior of f”(x) when f'(x) is decreasing, we need to consider a few possibilities:
1. If f'(x) is continuously decreasing, it implies that the rate of change of the slope is consistently decreasing. In this case, f”(x) is negative. This means that the function f(x) is concave down, and its graph takes on a “U” shape.
2. If f'(x) starts decreasing, then becomes constant, f”(x) will be zero. This means that the function f(x) has a point of inflection where the concavity changes.
3. If f'(x) starts decreasing, then becomes increasing again, f”(x) will be positive. This indicates that the function f(x) is concave up, and its graph takes on a “∪” shape.
In summary, if f'(x) is decreasing, f”(x) can be negative, zero, or positive, depending on the behavior of the derivative.
More Answers:
Understanding Decreasing Functions and their Derivatives in Calculus | ExplainedUnderstanding Concavity in Math | Explaining the Relationship Between a Function’s Curvature and its Second Derivative
Understanding Concave Down Functions | Relationship between Graph Shape and Second Derivative