Critical points ?Find the critical points of the following functions on the given intervals. Identify the absolute minimum and absolute maximum values (if possible). Graph the 3 2 function to confirm your conclusions. f(x) = 2x ? 3x ? 36x + 12 (? ?,?)

Solution 7RE Step 1 In this problem we have to find the critical points of the function 3 2 f(x) = 2x 3x 36x+12 and also we have to identify absolute maximum and absolute minimum. First let us see the definitions of critical point, absolute maximum,absolute minimum. Critical point: An interior point cof the domain of a function f at which f (c) = 0or f(c)fails to exist is called a critical point of f Absolute maximum: The highest point over the entire domain of a function or relation is the absolute maximum. Absolute minimum: The lowest point over the entire domain of a function or relation is the absolute minimum. Step 2 Given f(x) = 2x 3x 36x+12 on (,) Since f(x) is a polynomial, its derivative exists everywhere. By the definition of critical points, If f has critical points they are points at whi(x) = 0. 2 Here f (x) = 6x 6x36 Now f (x) = 0 6x 6x36 = 0 6 (x x6) = 0 2 (x x6) = 0 Factorizing the above term we get, (x+2)(x3) = 0 x = 2, 3