Concavity and Point of Inflection We have seen that the first derivative when set equal to zero will give us the critical points of a function and thus we can find the intervals of increase and decrease. This allows us to identify local maximum and local minimum points. If y = f(x) has a critical point at x = c, with f'(c) = 0, then the behaviour of f(x) at x = c can be analyzed through the use of the by analyzing as follows: a. The graph is concave up, and r - e is the location of a local minimum value of the function, if f"(c) > 0. local minimum b. The graph is concave down. and x = c is the location of a local maximum value of the function. if f[c) = 0. local maximum c. Iff(c) = D, the nature of the critical point cannot be determined without further work. A occurs at (c, f (c)) on the graph of y = f(x) if changes sign at x = c. That is, the curve changes from to or vice versa.Eg.1: Find the critical points for the following functions. Determine whether the functions have a local maximum or minimum. a) f(x)=x' -6x2 -15x+10 b) f(x)=2x3 -10x+3 Eg.2: Discuss the function y = X x' +1 with respect to the concavity and points of inflection