• The graph of f is blue.
  • The graph of the derivative of f is red.
  • The graph of the second derivative of f is brown.
  • The graph of the tangent line at (c,f(c)) is purple.
  • The graph of the line segment between (a,f(a)) and (b,f(b)) is black.

In the given example, f(x) = (1 - x2)1/2, we see that the derivative (red graph) is decreasing and the second derivative (brown graph) is strictly negative. Hence, f is concave downwards everywhere. In the second graph, we see that the tangent line (purple graph) at (c, f(c)) is above the graph of f except for the point (c, f(c)). The line segment between (a,f(a)) and (b,f(b)) (black line) lies below the graph of f except for the endpoints. Change the values of c, a and b and check the position of the new tangent line and line segment.