Locating roots and iterationEdexcel A-Level Maths: Mind map

Exam questions on Locating roots and iteration

1. Let $f(x)=x^3-3x-5$, a continuous function. The equation $f(x)=0$ has a single real root $\alpha$.
   Show that $\alpha$ lies in the interval $[2.2,\,2.3]$.2 marks
2. The root $\alpha$ of $x^3-3x-5=0$ is estimated using the iteration $x_{n+1}=\sqrt[3]{3x_n+5}$ with $x_0=2$.
   Find $x_3$ and $x_4$, and hence write down the value of $\alpha$ correct to 2 decimal places.2 marks
3. Let $f(x)=\frac{1}{x-2}$ for $x\ne2$, and $g(x)=x^2-5x+6$.
   Show that $f(1)$ and $f(3)$ have opposite signs, and explain why this does not show that $f(x)=0$ has a root in $[1,3]$.3 marks