Location of rootsEdexcel International A Level Maths: Mind map

Exam questions on Location of roots

1. The function $f$ is defined by $f(x)=x^3+x-5$ for $x\in\mathbb{R}$. The equation $f(x)=0$ has exactly one real root $\alpha$.
   Given that $f(1.6)=0.696$, explain how the values of $f(1.5)$ and $f(1.6)$ show that $1.5<\alpha<1.6$.2 marks
2. The functions $h$ and $p$ are defined by $h(x)=\frac{1}{x-2}$ for $x\neq2$, and $p(x)=x^2-2$ for $x\in\mathbb{R}$.
   A student notes that $h(1)<0$ and $h(3)>0$, and concludes that $h(x)=0$ has a root in $[1,3]$. Explain why this conclusion is wrong.2 marks
3. The function $f$ is defined by $f(x)=e^x-3x$ for $x\in\mathbb{R}$. The equation $f(x)=0$ has two real roots, $\alpha$ and $\beta$, with $\alpha<\beta$.
   Show that $\alpha$ lies in the interval $[0.5,0.7]$.3 marks