What are local maximum and minimum values of a function?

STEP_BY_STEP SOLUTION Step-1 Let f be defined on the interval [a,b] , and x be0the interior point on [a,b]. 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 ar0 all less than f(x ). 0 That is , f(x) f(x )0 Thus, the graph of f nea r x0 has a peak at x 0 A function f has a local minimum or relative minimum at a point x if th0 values f(x) of f for x ‘near’ x a0e all greater than f(x )0 . That is f(x) f(x 0. Thus, the graph of f near x h0s a trough at x . (T0 make the distinction clear, sometimes the ‘plain’ maximum and minimum are called absolute maximum and minimum). Step_2 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 o rdered list of special points in the interval: a=x < x < x ………………< x =b 0 1 2 Between each pair x < x + 1 of points in the list, choose an auxiliary point t + 1 i i Evaluate the derivative f at all the auxiliary points. For each critical point x , we have theiauxiliary points to each...