Show that the equation 2^x = 2 − x^2
has at least two real roots

- anonymous

Show that the equation 2^x = 2 − x^2
has at least two real roots

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- katieb

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- anonymous

Solve for 0, then factor.

- amistre64

use the determinate..i think its called of the quad formula.... if its 0 only 1 root, if it (+) 2 roots, it its (-) imaginary roots

- anonymous

but its 2^x an exponential fuction...so how to use quadratic here ?

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## More answers

- amistre64

thought that was a typo lol

- myininaya

its called the discriminant

- anonymous

Same here.

- amistre64

soooo close......

- myininaya

b^2-4ac

- anonymous

but how does discriminant help ?

- anonymous

\[-x^2+2^x -2 = 0\]

- amistre64

x^2 = -2^x +2
square root both sides?

- amistre64

x = +-sqrt(-2^x +2)

- anonymous

sry but i m still in a fix ....jeezzzzz

- amistre64

that doesnt tell you the roots, but it might show that there are 2 of them :)

- anonymous

ahh..got ya !!!

- anonymous

hey but the roots can be complex by taking the roots???

- anonymous

by taking the square root

- amistre64

roots can be complex and most are... but the only one to concern yourself with are the "real" roots

- anonymous

yes...when it was +- sqrt( 2- 2^x). that does not show that v r gonna get two real roots which is the question asking for?

- amistre64

if you wanna limit your self to even values of x

- anonymous

but there is no domain given in the question

- anonymous

meanwhile
i checked on wolfram alpha

- amistre64

oh wait... is it (-2)^x or -(2^x)??

- anonymous

they had a graph which cuts on two points

- anonymous

the latter one

- amistre64

maybe log rules would help

- anonymous

would you know guys what is uniform continuity?

- amistre64

log(2^x) = log(2-x^2)
x log(2) = log(2-x^2)
log(2) = log(2-x^2)^1/x
2 =xroot(2-x^2) ??

- amistre64

uniform?... id have to look it up

- anonymous

ok

- anonymous

f(x) = sin x is continuous on R. Is it
uniformly continuous on R?

- amistre64

id vote yes, its periodic, repetitive, and constantly the "same"

- amistre64

if "x" is a fraction that gets us back to integer exponents with the first question....

- anonymous

http://www.wolframalpha.com/input/?i=2^x+%3D+2+%E2%88%92+x^2 check out that one for me plsssssss

- amistre64

..... yeah.....thats correct :)

- anonymous

so u think graph would be enough?

- amistre64

the graph itself is visual proof, the mathematical proof seems a but difficult to reach.

- anonymous

yes...and the second question like you said or is there any better way?

- amistre64

if you graph the sin function, it sounds as if it would fit a "uniform" continuity.

- anonymous

yes...thank you.... can i ask you anything else or i m just pissing u off :(

- amistre64

lol.... you can ask....i might not have an answer tho :)

- anonymous

how to write pi as a symbol on computer?

- amistre64

on any computer? or in here?

- anonymous

any computer

- amistre64

its the ascii code 227 i think, there should also be a character keyboard in your menu that will bring up a virtual keyboard to find it on

- anonymous

yeah mayb in microsoft word ...they might have it ...thanx

- amistre64

programs..accessories....system tools....character maap is where mine is

- anonymous

hey let me check that one ...thanx

- amistre64

П lol

- anonymous

hey u got that...i might copy it n save it on word ...lol

- anonymous

ППППППППППП hhahah

- amistre64

Пп hehehe

- anonymous

lim x->pi/3 ((3+2cosx)^1/2 - 2) / ln (1+sin3x) ??? whats ur views on that one?

- amistre64

pi/3 = 60 degrees.....
is [sqrt(3+2cos(x)) - 2] / ln(1+sin(3x)) right?

- anonymous

exactly

- amistre64

sqrt(3+2cos(60)) - 2 sqrt(3 + 2(1/2)) - 2
------------------- = -----------------
ln(1+sin(180)) ln(1+0)

- anonymous

but thats a limit question ...plugging in the values making it 0/0

- amistre64

sqrt(4) - 2 0
--------- = ---
ln(1) 0
I see the dellima

- amistre64

the derivative of each would be....

- amistre64

Dx(ln(x)) = 1/x
Dx(ln(1+sin(3x))) = 3cos(3x)/1+sin(3x)
the reciprocal of that is what youll need

- amistre64

Dx(sqrt(3+2cos(x))-2) = -sin(x)/sqrt(3+2cos(x))

- amistre64

-sin(x) (1+sin(3x))
----------------------
sqrt(3+2cos(x)) (3cos(3x)

- anonymous

yes i was also somewhere there

- amistre64

-sqrt(3) sqrt(3)
--------- = -------- is what I get
2(3)(-1)(2) 12

- anonymous

i have got 1/4sqrt3

- amistre64

same thing, rationalize your denominator

- amistre64

sqrt(3)/4(3)

- anonymous

got ya !!! u must b a teacher for sure ... aren't u ?

- amistre64

lol ..... nope, just an unemployed carpenter

- anonymous

wtf??? impossible ...to me u r a GREAT TEACHER !!!! and more of a great human being

- amistre64

thanx :)

- anonymous

u have been so helpful...these questions were all for my next week homwork assignment...now i m so relaxed...thanx bro....really appreciate ur help

- anonymous

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