Locating roots by change of signAQA A-Level Maths: Mind map

Exam questions on Locating roots by change of sign

1. The function $f(x)=x^3-5x+3$ is continuous for all real $x$.
   Show that $f(x)=0$ has a root between $x=0.6$ and $x=0.7$.2 marks
2. Two functions are defined for $x\ne1$ by $f(x)=\frac{1}{x-1}$ and for all real $x$ by $g(x)=(x-1)^2$.
   Explain why the change of sign of $f$ between $x=0$ and $x=2$ does not show that $f(x)=0$ has a root in that interval.2 marks
3. The equation $e^x=3x$ is written as $f(x)=0$, where $f(x)=e^x-3x$.
   Show that $f(x)=0$ has a root $\alpha$ in the interval $0.6<x<0.7$.3 marks