Let \( y=f(x) \) be the real valued function defined on the interval \( (0,\infty) \), satisfying \( y(1)=0 \) and the differential equation \( x\,\dfrac{dy}{dx}=y-x^3 \). Then which of the following statements is (are) TRUE?
The function \( f \) has a local minimum at \( x=\dfrac{1}{\sqrt{3}} \)
The function \( f \) has a local maximum at \( x=\dfrac{1}{\sqrt{3}} \)
The function \( f \) is increasing in the interval \( (1,\,2) \)
If \( g(x)=4x^3-5x^2+\dfrac{3}{2}x \) for \( x>0 \), then the number of elements in the set \( \{\,x\in(0,\infty):\,f(x)=g(x)\,\} \) is \( 2 \)
