In the given example, f(x) = x^{4} + x^{2} - 2, we see that the derivative (red graph) is increasing and the second derivative (brown graph) is strictly positive. Hence, f is concave upwards everywhere. In the second graph, we see that the tangent line (purple graph) at (c, f(c)) is below 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 above 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. |