suppose we we have following equations and conditions
Let $k$ be the number of real solutions of the equation
$e^x+x-2=0$ in the interval $[0, 1]$ and and let $n$ be the number of real solutions that are not in $[0,1]$ Which of the following is true?
$k=0$ and $n=1$
$k=1$ and $n=0$
$n=k=1$
$k>1$
$n>1$
first of all what i have tried is this:if we differentiate we get following thing
$e^x=-1$
but how it is possible?,using wolfram alpha i got this result
http://www.wolframalpha.com/input/?i=e%5Ex%2Bx-2%3D0
but how to prove it using mathematical procedure?have to i use newtons method for compute actual root or there is some specific theorem which helps me to determine it more easily?
Best Answer
Okay, I am moving my comment as an answer:
Use derivative test to see if the function is increasing or decreasing. A strictly increasing/ decreasing function must be injective, so it can have at most one zero. Note that the function $e^x + x - 2$ takes both positive and negative values by evaluating at 0 and 1. What can you conclude by the intermediate value theorem?
Also, you should not try to find extremum points to solve this question, for the zeroes of $e^x + x -2$ need not be extremum points.