Quick questions on Numerical methods: change of sign, iteration and Newton-Raphson - OCR A-Level Maths A

If $f$ is continuous and $f(a)$ and $f(b)$ have opposite signs, then $f$ has at least one root between $a$ and $b$. You must state continuity for the argument to be valid.

Newton-Raphson uses the tangent at the current estimate to leap to a better one. Geometrically, you follow the tangent line at $x_n$ down to the $x$-axis, and that crossing is the next estimate. When it works it roughly doubles the number of correct digits each step, so it usually converges very fast from a good starting value.

Show that $x^3 + x - 5 = 0$ has a root between $1$ and $2$. [2 marks]

With $f(x) = x^2 - 5$ and $x_0 = 2$, apply Newton-Raphson once. [3 marks]