# root of equation

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

am not sure what to do with the $lnx$ function
• Apr 11th 2010, 09:50 AM
You want the zeros of $f(x) = \ln x + x -4$, $0. 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 would i show that the equation $\ln x \:=\: 4-x$ has one real root?
Graph the functions: . $\begin{Bmatrix}y &=& \ln x \\ y &=& 4-x \end{Bmatrix}$ . and note their intersection(s), if any.
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