Drag on the graph to move the tangent point — see how the slope changes
The derivative f'(x) gives the slope of the tangent line. Where f'(x)=0, the function has a critical point (possible max/min). The second derivative f''(x) tells us about concavity: f''>0 → concave up (cup shape), f''<0 → concave down (cap shape). Where f''=0 is a potential inflection point where concavity changes. Drag on the graph or use the slider to explore!