Locating critical points a. Find the critical points of the following functions on the domain or on the given interval. b. Use a graphing unity In determine whether each critical point corresponds to a local minimum. local maximum. or neither. x ?x f(x) = e +e 2

STEP_BY_STEP SOLUTION Step-1 Let f be a continuous function defined on an open interval containing a number ācā.The number ācā is critical value ( or critical number ). If f (c) = 0 1 1 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Example ; Find all critical points of f(x) = x 8x . 2 Because f (x) is a polynomial function, its domain is all real numbers. 1 f (x) = 4x 16x f (x) = 0 =4x 16x 4x ( x - 4) = 0 4x(x+2)(x-2) = 0 , since a - b = (a+b)(a-b) X = 0 , x = -2 and x=2. 4 2 At x= -2 , then f(-2) = (2) 8(2) = 16 - 32 = -16. At x= 0 , then f(0) = (0) 8(0) = 0 - 0 = 0. At x= 2 , then f(2) = (2) 8(2) = 16 - 32 = -16. Hence, the critical points of f(x) are (2,16), (0,0), and (2,16). Step-2 e +e x a). The given equation is ; f(x) = 2 , here f(x) is exponential function , its domain is all real numbers .so, f(x) is continuous for all of x. e +ex Now , f(x) = 2 then differentiate the function both sides with respect to x. x x So , f (x) = d ( e +e ) dx 2...