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. 3 ?x f(x) = x e , on [-1,5]

Solution 53E Step-1 Critical value definition; Let f be a continuous function defined on an open interval containing a number 1 1 ‘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 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 fu nction with domain D and let c be a fixed constant in D . hen 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 3 x a). Thegiven function is f(x) = x e on [-1,5] .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. Now , f(x) = x e 3 x then differentiate the function both sides with respect to x. 3 dx f(x) = dx(x e x ) 1 3 d x x d 3 d d(v) d(u) f (x) = x dx (e )+ e dx (x ), since dx(uv) = u dx +v dx 3 2 =x ( (1) e x )+ e x/2 (3x ) 3 x x/2 2 3 x 2 x = (-x e )+ e (3x ) = (-x e )+ (3x e ) 2 = x e x ( -x +3) 1 2 c Since , from the definition f (c)=0 = c e ( - c+3) 2 c e c ( - c+3)...