Analytical Applications of Differentiation · Challenge problem

A company's profit $P(x)$ (in thousands of dollars) after $x$ months is modeled by $P(x)=x^3-6x^2+9x+k$ for a constant $k$. The profit is exactly $\$5{,}000$ (that is, $P(1)=5$) after $1$ month. At what other time $x>1$ does the profit return to exactly that same $\$5{,}000$ level?