Question

Show that if f(x) is a differentiable function with f(x)<0 for all x E R and with a local maximum at x=c, then g(x)=[f(x)]^2 has a local minimum at x=c?

Answer 1

if f(c) is a max then f'(c)=0 and the 1st d-test would give us the just to the left of c
f' would be positive and just to the right of c f' would be negative

g'(x)=2f(x)f'(x)
g'(c)=2f(c)f'(c)=2f(c)*0=0

just to the left of c g' will be negative since f<0 and f'>0
just to the right of c g' will be positive since f<0 and f'<0

thus g(c) is a min

Answer 2

how is f'(c)=0?

Answer 3

http://en.wikipedia.org/wiki/Fermat%27s_theorem_%28stationary_points%29

Answer 4

Zarkon, would this also suffice? (im trying to get the hang of Analysis)
f(c) being the max means:
\[f(x)\leq f(c)<0 \]
for all x in whatever interval we are talking about
By squaring both sides we obtain:
\[f(x) \leq f(c) 0 \Rightarrow 0 < (f(c))^2 \leq (f(x))^2\]

Answer 5

oops, that should be f(c) < 0 on the left side.

Answer 6

yes

Answer 7

cool. im gonna look at yours in more detail though.

Answer 8

you would need to specify a neighborhood though (not for all x)
this is a local max

Answer 9

ah i gotcha. and i like your proof better, very elegant :)

Answer 10

ok i see it ;)

Answer 11

@ joe : have you taken real analysis yet?

Answer 12

ive taken one semester in it, but tbh, i dont feel like i learned anything =/

Answer 13

i read the book every now and then (when i can pry myself away from my Linear Algebra books lol)

Answer 14

Thanks guys, but can you go through more word problems that I have posted. It will be great! =]

Answer 15

it takes a while for it all to sink in. My first course in RA kicked my butt..after about the 5th class I was ok ;)

Answer 16

Wow i dont even know if my school offers that many courses o.O It must be good though. I have a friend in graduate school that swears by Real Analysis lol

Answer 17

it gets neat when you learn measure theory and functional analysis

Answer 18

I had 1 class in RA as an undergrad and 5 as a grad

Answer 19

oh, i never thought of looking in the grad catalog. Maybe after RA II i'll see what grad classes are available.

Answer 20

you should ... it's good stuff

Answer 21

i cant believe joe looked at this problem
he doesnt like cal

Answer 22

its not cal though! (not really) >.< and notice i didnt use Cal to solve it :P

Answer 23

i guess lol

Answer 24

i guess it is kinda a cal problem lol

Answer 25

but i stick by my non-cal solution! rawr.

Answer 26

i would had done it zarkon's way
yours is pretty too though

Answer 27

my way isnt as interesting though. I didnt use all the facts presented (which sorta makes me feel like its wrong)

Answer 28

you don't need all the facts

Answer 29

that makes me feel better lol

Answer 30

your way works even if the function is not differentiable at c

Answer 31

im not feeling too good >.< i think i just need some food. I'll see you guys later :)

Answer 32

later

Answer 33

poor joe