Calc Theorems

extreme value theorem

in a function f is continuous of a closed interval [a,b], then f has an absolute max and absolute min on [a,b]

condition for a local max/min

if a function f has a local max/min at a number c, then either f’( c ) = 0 or f’( c ) DNE

rolle’s theorem

suppose f is a function defined on [a,b]. if

• f is continuous on [a,b]

• f is differentiable on (a,b)

• f(a) = f(b)

then there is at least one number c in (a,b) for which f’( c )=0

mean value theorem

let f be a function defined on [a,b]. if

• f is continuous on [a,b]

• f is differentiable on (a,b)

then there is at least one number c in (a,b) for which f’( c ) = f(b) -f(a) / b-athe average rate of change of f over [a,b]

1st derivative test

suppose f is a function that is continuous on an interval I. suppose c is a critical number of f and (a,b) is an open interval in I containing c

• if f’( c) > 0 for a<x<c and f’( c)<0 for c<x<b, then f( c) is a LOCAL MAX

• if f’( c) < 0 for a<x<c and f’( c)>0 for c<x<b, then f( c) is a LOCAL MIN

  • if f’( c) has the same sign on both sides of c, then f( c) is NEITHER a local max nor min

test for concavity

suppose f is a function continuous on a closed interval [a,b]. suppose f’ and f’’ exist on (a,b). if

• f’’(x)>0 on (a,b), f is CONCAVE UP

• f’’(x)<0 on (a,b), f is CONCAVE DOWN

condition for inflection point

suppose f is a differentiable function of (a,b) containing c. if (c,f( c)) is an inflection point of f, then f’’( c)=0 or DNE at c

2nd derivative test

suppose f is function for which f’ and f’’ exist on (a,b). suppose c lies in (a,b) and is a critical number of of.

• if f’’( c) <0, then f( c) is a LOCAL MAX

• if f’’( c)>0, then f( c) is a LOCAL MIN