Question

Suppose f '' is continuous on (-infinity, infinity).
If f '(-1) = 0 and f ''(-1) = -1,what can you say about f ?
a)At x = -1, f has a local maximum.
b)At x = -1, f has a local minimum.
c) At x = -1, f has neither a maximum nor a minimum.
d)More information is needed to determine if f has a maximum or minimum at x = -1.

If f '(4) = 0 and f ''(4) = 0,what can you say about f ?
a)At x = 4, f has a local maximum.
b)At x = 4, f has a local minimum.
c)At x = 4, f has neither a maximum nor a minimum.
d)More information is needed to determine if f has a maximum or minimum at x = 4.

Answer 1

If f '(-1) = 0 and f ''(-1) = -1
since f'' is negative, f' is decreasing, it is also zero at x=-1, so
for x<-1 it must be positive and for x?-1 f' is negative
f'(-1)=0 so f has an extreme point there and hence
for x<-1 f must be increasing and for x?-1 f must be negative
so that is a maximum
b)At x = -1, f has a local minimum.

Answer 2

http://en.wikipedia.org/wiki/Second_derivative_test

so
d)More information is needed to determine if f has a maximum or minimum at x = 4.

Answer 3

thank you for your help. on WebAssign, the first answer was marked wrong. Can we go through your explanation again?

Answer 4

Oh, sorry that was a typo, I copied the wrong anser
it is
d)More information is needed to determine if f has a maximum or minimum at x = 4.

The funtion is increasing, then reaches a maximum, then decreases
sorry about that

Answer 5

oh that was the second question and D was right. Thank you again for that one!
I'm talking about the f'(-1)=0 and f''(-1)=-1
x=-1 is a minimum

Answer 6

ah, didn properly copy the answer,
a)At x = 4, f has a local maximum.

Answer 7

the previous answer was still on the clipboard

Answer 8

The funtion is increasing, then reaches a maximum, then decreases sorry about that
so A

Answer 9

ah ok. well thanks again for the help. I was stumped on these two questions. I was able to do the rest but i've spent a while not understanding this.

Answer 10

Do you get it now?

Answer 11

yeah i have the gist of it

Answer 12

the derivative gives whether a function is increasing (when positive) or decraesing (when negative)
at an extremum of f it changes sign
but if f' is also zero, it may or may not change sign and you need higher derivatives to see what is going on

Answer 13

since f'' is negative then it's concave down yes? so the frown means x=-1 is a local max

Answer 14

for f

Answer 15

if f'' is negative it means that
f' is decreasing and zero so before the extremum
first postitive then negative
so f is at a maximum (first increasing then decreasing)

Answer 16

ah ok. I went over it again and I can officially say I understand it now. Thank you.

Answer 17

Welcom

Answer 18

e