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Example: Suppose f '(x) = x2 + 5x - 6= (x -1)(x + 6). (a) Identify the intervals of x-values on which f is increasing, decreasing, concave up and concave down. (b) Locate and identify where the local maximum point, the local minimum point, and any inflection poir occur . First, fund the critical points. Solve : f' ( x ) = 0 x-+ 5x-6 = 0 ( x - 1 ) ( * + 6 ) = 0 X = 1, x = - 6 : critical points Second, make a table of intervals of real numbers Intervals f' ( ) = x 7 + 5 x - 6 = ( x-1 ) ( X +6 ) Comments (-00,-6) f' ( 7 ) = (- 7-1 )( - 7+6)=(8/(-) 20 f increasing X = - 6 f' (- 6) = 0 : critical point - has a local max at x=- 6 ( - 6, 1 ) f' (0)= 0+0-6= -640 of decreasing X = 1 f' ( 1 ) = 0: critical point (1, 00 ) f has a local mun at x= 1 f'(10)=(10)+5(10)- 6 = 14470 + increasing Left to student to frush problem: determining where f is concave up, concave down and location of inflection pout . f' (x) = x+5x-6 = (x-1)(x+6)