Functions
Exponential-cubic intersection — counting solutions, paragraph II
MJAT_TS4_P2
Grade 12
Question:
**Paragraph II:**
Consider the equation $ae^x = x^3$.
**Question:** The number of real solutions when $a\leq 0$ is $\beta$, and when $a=\dfrac{27}{e^3}$ is $\gamma$. Then $\beta+\gamma=$
Step-by-Step Solution
Key Concept: For $a=0$: $0=x^3\Rightarrow x=0$, one solution. For $a<0$: $ae^x<0$ and $x^3<0$ for $x<0$. There is exactly one intersection for $x<0$. So $\beta=1$ for $a<0$ and $\beta=1$ for $a=0$: total $\beta=1$. For $a=27/e^3$: the curves $ae^x$ and $x^3$ are tangent (touching at exactly one point, $x=3$), so $\gamma=1$. $\beta+\gamma=1+1=2$.
$\beta=1$, $\gamma=1$. $\beta+\gamma=\mathbf{2}$.
Correct Answer: 2