Real roots of $f(x) = 5 \sqrt x + 37[ \sqrt[27] x] = 1$ in $[0,1]$

Question

Find real roots of the following polynomial which belong to the closed interval $[0,1]$ :

$$f(x) = 5 \sqrt x + 37[ \sqrt[27] x] - 1$$

How to approach such a problem? I don't know where to start.

Answer 1

As a start, if $f(x) =5\sqrt{x} + 37x^{1/27} - 1 $, $f(0) = -1$ and $f(1) = 41 $, so there is a root there.

If $x = 1/25$, $5\sqrt{x} = 1$ so $f(1/25) > 0$, so the root is in $(0, 1/25)$.

Looking at the other term, if $x = (1/37)^{27} \approx 4.56\cdot 10^{-43} $, $f((1/37)^{27}) > 0$ so the root is between $0$ and this.

Now apply Newton or some other iteration to get a more accurate estimate.