# root of equation

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• Apr 11th 2010, 09:31 AM
sigma1
root of equation
hello how would i show that the equation $\displaystyle lnx = 4-x$ has one real root..

am not sure what to do with the $\displaystyle lnx$ function
• Apr 11th 2010, 09:50 AM
maddas
You want the zeros of $\displaystyle f(x) = \ln x + x -4$, $\displaystyle 0<x<+\infty$. Observe by differentiation that f is strictly increasing. Therefore if f has at most one root, since if a<b were two distinct roots we could chuse a<c<b, and 0=f(a)<f(c)<f(b)=0, a contradiction. To see that one root exists, evaluate f near zero to get a negative value and then above 4 to get a positive value and use the IVT.
• Apr 11th 2010, 10:12 AM
Soroban
Hello, sigma1!

How about a graphing approach?

Quote:

How would i show that the equation $\displaystyle \ln x \:=\: 4-x$ has one real root?

Graph the functions: .$\displaystyle \begin{Bmatrix}y &=& \ln x \\ y &=& 4-x \end{Bmatrix}$ . and note their intersection(s), if any.
Code:

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