Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The function f(x) = ? x has a local maximum on the interval [0, 1]. b. If a function has an absolute maximum, then the function must be continuous on a closed interval. c. A function ?f? has the property that ?f??(2) = 0. Therefore, ?f? has a local maximum or minimum at x ? ? = 2. d. Absolute extreme values on an interval always occur at a critical point or an endpoint of the interval. e. A function ?f? has the property that ?f??(3) does not exist. Therefore, ?x? = 3 is a critical point of ?f?.

STEP_BY_STEP SOLUTION Step-1 Critical point definition; 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) = 0 1 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 f nction w ith domain D and let be a fixed constant in D . Then the output value f ) 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 Local maximum , local minimum definition; Let f be defined on the interval [a,b] , and x be the interior point on [a,b]. 0 A function f has a local maximum or relative maximum at a point x 0 if the values f(x) of f for x ‘near’ x are 0ll less than f(x ). 0 That is , f(x) f(x ) 0 Thus, the graph of f near x has a p eak at x . 0 0 A function f has a local minimum or relative minimum at a point x if the 0 values f(x) of f for x ‘near’ x are all gre0ter than f(x ) . 0 That is f(x) f(x ). 0 Thus, the graph of f near x has a trou0h at x . (To make the distin0tion clear, sometimes the ‘plain’ maximum and minimum are called absolute maximum and minimum). To find the local maxima and minima of a function f on an interval [a,b] ; Solve f(x) = 0,to find c ritical points of f. Drop from the list any critical points that aren't in the interval [a,b]. Add to the list the endpoints (and any points of discontinuity or non-differentiability): we have an...