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+ l)4/3 on[?8, 8]

STEP_BY_STEP SOLUTION Step-1 Let f be a continuous function defined on an open interval containing a number ‘c’.The number ‘c’ is critical value ( or critical number ). If f (c) = 0 1 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 + 1) 4/3 , on [-8,8].Clearly the function is 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 + 1) 4/3then differentiate the function both sides with respect to x. f (x) = dx (x + 1)4/3 = 4 (x + 1) (4/3)1 ,since d ( x ) = nx n1 and d (c) = 0, where c is 3 dx dx constant. 4 1/3 = 3 (x + 1) Since , from the definition f (c)=0 = 4 (c + 1)1/3...