(i) Show that the equation

$$\epsilon x^{4}-x^{2}+5 x-6=0, \quad|\epsilon| \ll 1$$

has roots in the neighbourhood of $x=2$ and $x=3$. Find the first two terms of an expansion in $\epsilon$ for each of these roots.

Find a suitable series expansion for the other two roots and calculate the first two terms in each case.

(ii) Describe, giving reasons for the steps taken, how the leading-order approximation for $\lambda \gg 1$ to an integral of the form

$$I(\lambda) \equiv \int_{A}^{B} f(t) e^{i \lambda g(t)} d t$$

where $\lambda$ and $g$ are real, may be found by the method of stationary phase. Consider the cases where (a) $g^{\prime}(t)$ has one simple zero at $t=t_{0}$ with $A<t_{0}<B$; (b) $g^{\prime}(t)$ has more than one simple zero in $A<t<B$; and (c) $g^{\prime}(t)$ has only a simple zero at $t=B$. What is the order of magnitude of $I(\lambda)$ if $g^{\prime}(t)$ is non-zero for $A \leq t \leq B$ ?

Use the method of stationary phase to find the leading-order approximation to

$$J(\lambda) \equiv \int_{0}^{1} \sin \left[\lambda\left(2 t^{4}-t\right)\right] d t$$

for $\lambda \gg 1$.

[You may use the fact that $\int_{-\infty}^{\infty} e^{i u^{2}} d u=\sqrt{\pi} e^{i \pi / 4}$.]