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 function to confirm your conclusions. f(x) = x (9 ? x ) on [? 4,4]

Solution Step 1 In this problem we have to find the critical points of the function f(x) = 2x ln x +10and 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) = x (9x ) on [4,4] 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 which f (x) = 0. Here f (x) = x1 31(9x )+x (2x) 3 1 2 2 4 f (x) = x3(9x )2x 3 4 32 x3 3 f (x) = 3x 3 2x Now f (x) = 0 4 3x 3 x3 2x = 0 3 9x 32x 6x = 0 3 2 Multiply throughout by x we get, 2 2 9 x 6x = 0 97x = 0 2 9 x = 7 (Since ln and ecancel each other) x = ± 3 = ± 1.1339 7 Step 4 Thus the critical points are x = 1.1339 and x = 1.1339which lie in the interval [4,4] . These critical points and the end points help us to locate the absolute extrema. Now let us find the value of f(x)at these points. 1 At x = 4, f(4) = (4) (9(4) ) = (1.5874)(916) = 11.1118 At x = 1.1339, 3 2 f(1.1339) = (1.1339) (9(1.1339) ) = (1.04277)(91.2857) = 8.0442 1 At x = 1.1339, f(1.1339) = (1.1339) (9(1.1339) ) = (1.04277)(91.2857) = 8.0442 1 At x = 4, f(4) = (4) (9(4) ) = (1.5874)(916) = 11.1118