1. **State the problem:** We are given the function $F(x) = x^3 - 12x^2 + 36x$ and we want to analyze or solve it. 2. **Formula and rules:** This is a cubic polynomial function. To find its roots, we set $F(x) = 0$ and solve for $x$. 3. **Set the equation:** $$x^3 - 12x^2 + 36x = 0$$ 4. **Factor the polynomial:** First, factor out the greatest common factor $x$: $$x(x^2 - 12x + 36) = 0$$ 5. **Solve each factor:** - From $x = 0$, one root is $x = 0$. - For the quadratic $x^2 - 12x + 36 = 0$, use the quadratic formula or factorization. 6. **Factor the quadratic:** $$x^2 - 12x + 36 = (x - 6)^2$$ 7. **Solve the quadratic:** $$(x - 6)^2 = 0 \\ x - 6 = 0 \\ x = 6$$ 8. **Roots of the function:** The roots are $x = 0$ and $x = 6$ (with multiplicity 2). 9. **Summary:** The function $F(x) = x^3 - 12x^2 + 36x$ has roots at $x = 0$ and $x = 6$ (double root).