use the intermediate value theorem to show that there is a root of the given equation in the specified interval.

Question

use the intermediate value theorem to show that there is a root of the given equation in the specified interval.

 

ln x = x - √x, (2,3)

Answer 1

Let f(x) = lnx - x + √x

 

f is continuous on the interval [2,3].

 

f(2) = ln2 - 2 + √2  > 0

 

f(3) = ln3 - 3 + √3  < 0

 

By the Intermedite value Theorem, there is at least one number, c, between 2 and 3 such that f(c) = 0.

 

So, lnc - c + √c = 0

 

Therefore, lnc = c + √c