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) = xe ?x/2 on [0,5]

TEP_BY_STEP SOLUTION Step-1 Critical value definition; 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) = 01 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 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) = xex/2on [0,5] . Clearly the function is a exponential function and it is continuous for all of x . Now , we have to find out the critical points of f on the given interval. x/2 Now , f(x) = xe then differentiate the function both sides with respect to x. d d x/2 dxf(x) = dx (xe ) f (x) = x d (ex/2 )+ e x/2 d (x), since d (uv) = u d(v)+v d(u) dx dx dx dx dx = x( 1 ex/2 )+ e x/2(1) 2 x x/2 x/2 x/2 x = 2 e + e = e ( 2 +1) 1 c/2 c Since...