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. 2 ? f(x) = x 1/2( ?4) 0n [0,4] 5

Solution 50E 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) = 0 1 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Step-2 Absolute extreme values 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 function w ith 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 1/2 x2 a). The given function is f(x) = x (54) 0n [0,4] .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. 2 5/2 Now , f(x) = x 1/2 (54) = ( x5 4x 1/2) , for the critical points , we have to differentiate the function both sides with respect to x. d d x5/2 1/2 dx f (x) = dx ( 5 4x ) 1 d x5/2 d 1/2 d d f (x) = dx ( 5 ) (dx ),since dx ( u-v) = dx ( u) - dx = ( )( )( x (5/2)1) - 4 ( )x (1/2)) , since d ( x ) = nx 5 2 2 dx = 2 ( x ) - 2x 1/2) 1 1 3/2 1/2 Since , from...