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) = cos (x) on [0,?]

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) = 01 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) = cos (x) , on [0 , ].Clearly the function is a trigonometric function and it is continuous for all of x , and also the function is even function it gives only positive values. Now , we have to find out the critical points of f on the given interval. 2 Now , f(x) = cos (x) then differentiate the function both sides with respect to x. 1 d 2 f (x) = dx(cos (x)) d = 2cos(x) dx(cos(x)) , [ since dx( x ) = nx n1 and dx (cos (x )) = -sin( x )n dx ( x )] = -2cos(x) sin(x) = -sin(2x) , since by the formula Since ,...