Baltic Way 1991 · Problem 9

Solution: Studying the graphs of the functions $a e^{x}$ and $x^{3}$ it is easy to see that the equation always has one solution if $a < 0$ and can have $0$, $1$ or $2$ solutions if $a > 0$. Moreover, in the case $a > 0$ the number of solutions can only decrease as $a$ increases and we have exactly one positive value of $a$ for which the equation has one solution - this is the case when the graphs of $a e^{x}$ and $x^{3}$ are tangent to each other, i.e., there exists $x_{0}$ such that $a e^{x_{0}} = x_{0}^{3}$ and $a e^{x_{0}} = 3 x_{0}^{2}$. From these two equations we get $x_{0} = 3$ and $a = \frac{27}{e^{3}}$. Summarizing: the equation $a e^{x} = x^{3}$ has one solution for $a \leq 0$ and $a = \frac{27}{e^{3}}$, two solutions for $0 < a < \frac{27}{e^{3}}$ and no solutions for $a > \frac{27}{e^{3}}$.