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. ? ? ? ? f(?x)=2x? sin ?x;[?2, 6]

Solution 49E Step-1 Critical point 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 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 be a fun ction wit h domain D and let c be a fixed constant in D . Th 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 x a). The given functio n is f(x) = 2 sin(x) on [ -2,6].Clearly the function contains both trigonometric, and algebraic functions 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) = 2x sin(x) for the critical values we have to differentiate the function both sides with respect to x. d (f(x)) = d (2 sin(x) ) dx dx 1 x x f (x) = sin(x) dx (2 ) +2 dxsin(x) , since dx(uv) = u dx(v)+v dx(u) x x d x x d = sin(x)(2 (ln(2) )+2 cos(x) , since dx (2 ) = 2 (ln(2)), dx sin(x) = cos(x). = 2 (ln(2))sin(x) +2 cos(X) = 2 (ln(2)sin(x)+cos(x)) 1 c Since , from the definition f (c)= 0 = 2 (ln(2)sin(c)+cos(c)) c 2 (ln(2)sin(c)+cos(c)) = 0 That is , 2 = 0 and (ln(2)sin(c)+cos(c)) = 0 . c So, 2 = 0 for this take log on both sides then c ln( 2 ) = ln(0) C (ln(2)) = ln(0), since ln( a ) = m(ln(a)) ln(0) C = ln(2) = undefined , since ln(0) is undefined. The second result...