Question

How do I show that equation `(5+ln(x))^sqrtx - x^3=0` has a root in interval `[2.2,3]`?

Answer 1

We see using a calculator:

`(5 + ln(3))^sqrt(3) - 3^3 = -4.088`

And

`(5 + ln(2.2))^sqrt(2.2) - (2.2)^3 = 2.875`

Since the function `f(x) = (5 + lnx)^sqrt(x) - x^3` is continuous on `[2.2, 3]`, it must cross the x-axis at some point during that interval. How else would it go from positive to negative?

This is called the intermediate value theorem.

Hopefully this helps!