A graph is concave up when its second derivative is positive. This means the function curves upward, and the slope (first derivative) is increasing over the interval. In simpler terms, if f′′(x)>0f''(x)>0f′′(x)>0, then the graph of f(x)f(x)f(x) is concave up.
Explanation of Concavity and Second Derivative
- The second derivative f′′(x)f''(x)f′′(x) measures the curvature of the function f(x)f(x)f(x).
- If f′′(x)>0f''(x)>0f′′(x)>0 on an interval, the graph is concave up there, resembling a "U" shape or an upward-facing bowl.
- The first derivative f′(x)f'(x)f′(x) is increasing when the graph is concave up.
- Conversely, if f′′(x)<0f''(x)<0f′′(x)<0, the graph is concave down (curves downward).
Summary
- Concave up ↔ second derivative positive (f′′(x)>0f''(x)>0f′′(x)>0)
- This indicates the function is bending upward and the slope is increasing.
This relationship helps in understanding the shape of functions, their minima and maxima, and is a key concept in calculus graph analysis.
