Explanation

(a) The potential energy of a particle is given by
$V(x) = \left( \frac{x^{4}}{4} - \frac{x^{2}}{2} \right)$
For minimum value of V,$\frac{dV}{dx} = 0$
$\therefore\ \frac{4x^{3}}{4} - \frac{2x}{2} = 0\ \Longrightarrow x = 0,x = \pm 1$
So, $V_{MIN}(x = \pm 1) = \frac{1}{4} - \frac{1}{2} = \frac{- 1}{4}J$
$\therefore\ \ \ K_{MAX} + V_{MIN}$=Total mechanical energy$\ $
$K_{MAX} = \left( \frac{1}{4} \right) + 2 \Longrightarrow K_{MAX} = \frac{9}{4}$
Or $\frac{mv^{2}}{2} = \frac{9}{4} \Rightarrow v = \frac{3}{\sqrt{2}}ms^{- 1}$