Applications Of Derivatives - II

If a <= x <= b and 0 in [a,b], then 0 <= |x| <= max(|a|,|b|).

If a <= x <= b and 0 not in [a,b], then min(|a|,|b|) <= |x| <= max(|a|,|b|).

Working rule on closed interval: compute f'(x), find critical points, evaluate f at critical points and end-points, then compare values.

f(c) >= f(x) for all x in (c-h, c+h), h>0

f(c) <= f(x) for all x in (c-h, c+h), h>0

f'(x)=0

f'(x)=0 or f'(x) does not exist

If f'(x)>0 just left of c and f'(x)<0 just right of c, then local maximum at x=c.

If f'(x)<0 just left of c and f'(x)>0 just right of c, then local minimum at x=c.

If f'(x) keeps the same sign on both sides of c, then neither local maximum nor local minimum; c is a point of inflexion.