Point of inflexion and concavity

The concept of increasing function, point of inflexion and concavity can be understood with an illustration.

 

Concavity

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.

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