The concept of increasing function, point of inflexion and concavity can be understood with an illustration.
a. Strictly increasing. f(x) is strictly increasing when f'(x) > 0. ( x<1.423 or x> 2.577) . These points also coincide with the maximum and minimum point.
b. Strictly decreasing. f(x) is strictly decreasing when f'(x) < 0. (1.423 < x < 2.577)
c. Point of inflexion is the point where the curve changes curvature or concavity. f'(x) does not change sign, and f” (x) = 0 and changes sign. In the above graph, the curve changes from convex to concave at x = 2.
d. Concave upwards. f(x) concave upwards when f”(x) > 0 or f'(x) is strictly increasing. f(x) concave upwards at x> 2, after f'(x) reaches a minimum, at the point of inflexion.
e. Concave downwards or convex. f(x) concave downwards when f”(x) < 0 or f'(x) is strictly decreasing. f(x) concave downwards for x < 2.