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. f?? ?)= ? ?x?2+1)2 on [?2, 2]

STEP_BY_STEP SOLUTION Step-1 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) = 0 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Step-2 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 function with domain D and let c be a fixed constant in D. Then the output value f(c) 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 , on [-2,2].Clearly the function is rational function and (x +1) 2 it is continuous for all of x , because the denominator is not equal to zero for all values of x. Now , we have to find out the critical points of f on the given interval. Now , f(x) = x 2 then differentiate the function both sides with respect to x. (x +1) 1 d x f (x) = dx 2 2 (x +1) ( (x +1) )ddxxdx (x +1) )2 d u vdxuuddx) = 2 4 , since ( ) = 2 (x +1) dx v v 2 2 2 d 2 ( (x +1) )(1)x 2( (x +1) dx(x ) d n n1 d = (x +1)4 , since dx ( x ) = nx dx (x). ( (x +1) )(1)x 2( (x +1) (2x) = 2 4 (x +1) 2 2 2 = (x +1) ( 2x +1) 4x ) (x +1)4 2 = 12 x 3 , ince cancel out the like terms. (x +1) 1 13 c2 Since , from the definition f (c)=0 = 2 3 (c +1) 13 c2 2 2 3 = 0 1- 3c = 0 (c +1) 2 2 3c = 1 c = Therefore, c = 1 and c = 1 3 3 1 1 Clearly 3 and 3 ies between [-2,2]....