Absolute maxima and minima a. Find the critical points off on the given interval. b. Determine the absolute extreme values off on the given interval. c. Use a graphing utility to confirm your conclusions.

Solution 55E Step-1 Critical value definition ; Let f be a continuous function defined on an open interval containing a 1 number ācā.The number ācā is critical value ( or critical number ). If f (c) = 0 or 1 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 a nction with domain D and let c b a fixed constant in D. The n 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 x a). The given function is f(x) = x4 on [6,12] . Clearly the function is a rational function and here the denominator is not equal to zero . So , x-4 > 0 That is , x > 4 Therefore , x belongs to (4 , ) Therefore, the given function is continuous on [6 , 12] Now , we have to find out the critical points of f on the given interval. x Now , f(x) = x4 for the critical points , we have to differentiate the function both sides with respect to x. f (x) = d ( x ) dx x4 d(x) d d(u) d(v) = (x4) dxxdx x4) , since d ( ) = v dxu dx ( x4)2 dx v v2 x d ( x4)(1)2x4dx( x4) d 1 = ( x4)2 , since dx (x) = 2x ( x4)2x41) 2(x4) x = x4 = (x4) x4 2x 8 x = (x4) x4...