Absolute maxima and minima a. Find the critical points of f on the given interval. b. Determine the absolute extreme values off on the given interval. c. Use a graphing utility to confirm your conclusions. 3 2 ? f(x) = x ? 2x ? 5x + 60n [4,8]

STEP_BY_STEP SOLUTION Step-1 Critical point definition; Let f be a continuous function defined on an open interval containing a 1 number ‘c’.The number ‘c’ is critical value ( or critical number ). If f (c) 1 = 0 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Step-2 Absolute extreme value definition; When an output value of a function is a maximum or a minimum over the entire domain of the function, the value is called the absolute maximum or the absolute minimum. Let f be afunction with domain D and let c be a fixed constant in D . Then the output value f( ) is the 1. Absolute maximum value of f on D if and only if f(x) f(c) , for all x in D. 2. Absolute minimum value of f on D if and only if f(c) f(x) , for all x in D. Step_3 a). The given function is f(x) = x 2x 5x + 6 ,0n [4,8] . Clearly the function is a polynomial function , and it is continuous for all of x. Now , we have to find out the critical points of f on the given interval. Now , f(x) = x 2x 5x + 6 t hen differentiate the function both sides with respect to x. 1 d 3 2 f (x) = dx ( x 2x 5x + 6) d 3 d 2 d d = dx( x ) dx(2x ) dx5x + dx 6 = 3 x - 2 (2 x) -5 (1) +0 = 3 x - (4 x) -5 since dx (x ) = n x n1 , and dx (Cx) = C dx(x), c is constant. 1 Since , from the...