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 side of it: ti< x < i + 1. i here are four cases b est remembered by drawing a picture!: if f( t i > 0 , and f(t + i ) < 0 . (so f is increasing to the left of the right of x , then i has a local maximum at x . 0 if f( t )i< 0 , and f(t + 1 i > 0(so f is decreasing to the left of x and increasing to the right of x ,then f has a local minimum at x . i 0 if f( t ) < 0 , and f(t + 1 ) < 0(so f is d ecreasing to the left of x and a i i decreasing to the right of x i then f has neither a local maximum nor a local minimum at x . 0 if f( t i > 0 , and f(t + 1 )i> 0(so f is i ncreasing to the left of x and a increasing to the right of x , f has n either a local maximum nor a local minimum at x i . The endpoints require separate treatment: There is the auxiliary point to just to the right of the left endpoint a, and the auxiliary point t just to the l endpoint b: At the l eft endpoint a, if f(t )< 0, (so f is d is a local maximum. At the l eft endpoint a, if f(t ) > 0, (so f is i is a local minimum. At the r ight endpoint b, if f(t ) < 0, (so f is d from the left) then b is a l At the r ight endpoint b, if f(t )>0 (so f is increasing as b is approached from the left) then b is a l