# Thread: help me with this intermediate theorem plz!!

1. ## help me with this intermediate theorem plz!!

starting from 0 degree latitude and proceeding in a westerly direction along the equator, let T(x) be the temperature at the point x at any given time. Assuming T(x) is a continuous function of x, show that at any fixed time. there are at least 2 diametrically opposite points on the equator, say a and a+180, that have exactly the same temperature.
Hint: construct a function F(x) using t(x) and use the intermediate value theorem.

all i can come up is that we need to prove:
T(x) = T(x+180)
which is T(x) - T(x+180) = 0

let f(x) = T(x) - T(x+180)

if we can prove that f(x) has a root, then T(x) can = T(x+180)

since T(x) and T(x+180) are continous function, so f(x) as well.

that is as far as i can go

how can i determine where f(x) is negative and positive ???

thanks!!!!

2. Originally Posted by dull1234
starting from 0 degree latitude and proceeding in a westerly direction along the equator, let T(x) be the temperature at the point x at any given time. Assuming T(x) is a continuous function of x, show that at any fixed time. there are at least 2 diametrically opposite points on the equator, say a and a+180, that have exactly the same temperature.
Hint: construct a function F(x) using t(x) and use the intermediate value theorem.

all i can come up is that we need to prove:
T(x) = T(x+180)
which is T(x) - T(x+180) = 0

let f(x) = T(x) - T(x+180)

if we can prove that f(x) has a root, then T(x) can = T(x+180)

since T(x) and T(x+180) are continous function, so f(x) as well.

that is as far as i can go

how can i determine where f(x) is negative and positive ???

thanks!!!!
You are off to a good start. Consider f(0) and f(180):

f(0) = T(0) - T(180)
f(180) = T(180) - T(360) = T(180) - T(0) = -f(0)

Think you can take it from there?

3. yep, thx. i figure that out when i am having dinner as well XD
but i sub in a and a + 180 instead of 0 and 180
basically just the same thing

so f(a) and -f(a) are two opposite number, since f(x) is a continous function, there should be a point within the interval [a, a+180] that makes f(x) = 0, which means T(x) = T(x+180)

XDXD
am i correct ?

4. That's right.

(You might want to consider the case f(a) = 0 separately, just to be sure you have all the possibilities covered. But what you have is correct.)