A particle of mass $m$ moves in a one-dimensional potential energy $U(x) = -ax^2 + bx^4,$ where $'a'$ and $'b'$ are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to
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$\frac{d V(x)}{d x}=\frac{d}{d x}\left(-a x^{2}+b x^{4}\right)$
$=-2 a x+4 b x^{3}=x\left(4 b x^{2}-2 a\right)=0$
$x_{1}=0$
$x_{2}=\sqrt{a / 2 b}$
$F=-k x$
$\frac{d F}{d x}=k=\frac{d^{2} V(x)}{d x^{2}}$
$k=\frac{d}{d x}\left(-2 a x+4 b x^{3}\right)=-2 a+12 b x^{2}$
$k(x=0)=-2 a$
$k(x=\sqrt{a / 2 b})=-2 a+6 a=4 a$
$\omega=\sqrt{\frac{k}{m}}$
$\omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{4 a}{m}}$
$\omega=2 \sqrt{\frac{a}{m}}$
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