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? ?) = sin 3?x? on [??/4, ?/3]

STEP_BY_STEP SOLUTION 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 b e a fixed constant in D . T en 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 fun ction is f(x) = sin(3x) on [ - , 4 3learly the function is a trigonometric 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) = sin(3x) for the critical values we have to differentiate the function both sides with respect to x. f (x) = d (sin(3x)) dx d d = cos(3x) dx (3x) , since dx sin(x) = cos(x). = 3cos(3x) , since d (cx) = c d (x) dx dx 1 Since , from the definition f (c)=0 =3 cos(3c) 3 cos(3c) = 0 cos(3c) =cos((2n+1) ), where n2= …-3,-2...0,1,2……………...