A graph is concave up on an interval if it "opens" upward, resembling a bowl or a big "U" shape. More precisely:
- All tangent lines to the graph on that interval lie below the curve.
- Equivalently, the slope of the tangent lines (the first derivative) is increasing on that interval.
- The second derivative of the function is positive over the interval.
In simpler terms, if the rate of change of the slope is increasing and the graph bends upward, it is concave up. This means visually the graph looks like it is curving upwards, and mathematically:
- f′′(x)>0f''(x)>0f′′(x)>0 (the second derivative is positive)
- f′(x)f'(x)f′(x) is increasing (the first derivative function slopes upward)
This concavity is separate from whether the function itself is increasing or decreasing; it only describes the curvature or "shape" of the graph, not its direction. Conversely, if the graph bends downward, the tangent lines lie above the graph and the second derivative is negative, the graph is concave down.
