Zeros of a Min Function

Let $f(x)=x^3+ax+\dfrac 14$ and $g(x)=-\ln x$.

1. For what value of $a$ is the $x$-axis tangent to the curve $y=f(x)$? 2. Writing $\min\{m,n\}$ for the smaller of $m,n$, let $h(x)=\min\{f(x),g(x)\}$ for $x>0$. Discuss the number of zeros of $h(x)$.