4.3-4.4 Graphing Functions con't

the second derivative

tells us where f(x) is concave up and concave down and helps us to find inflection points of f(x)

if f’’(x) < 0

f(x) is concave down (cant hold water)

if f’’(x) > 0

f(x) is concave up (can hold water)

inflection point

If f’’(x) changes sign at x= c and f(x) is continuous at x=c, then x=c is an __ of f(x)

if f’(c ) = 0 and f’’(c ) > 0 then

f(c ) is a local minimum U

if f’(c ) = 0 and f’’(c )< 0 then

f (c ) is a local maximum

if f’(c )=0 and f’’(c ) = 0

then it is inconclusive

steps

1. find intervals where f (x ) is concave up and concave down

   • find f’’(x)

   • find CPs of f’(x) (where f’’(x ) = 0/DNE)

   • construct sign chart

2. find x- coordinates of all inflection points of f(x)

   • where f’’(x) changes sign (will be the critical point)