Functions
Exponential-cubic intersection — number of solutions, paragraph II
MJAT_TS4_P2
Grade 12
Question:
**Paragraph II (continued):**
The number of real solutions of the equation $ae^x=x^3$ when $0<a<\dfrac{27}{e^3}$ is:
Step-by-Step Solution
Key Concept: For $0<a<27/e^3$: the line $ae^x$ (exponential curve) intersects $x^3$ in exactly 2 places (it's below the tangent level). For $x<0$: $ae^x>0$ but $x^3<0$ — no intersection. For $x>0$: the exponential grows slower than the cubic for large $x$, and starts below (at $x=0$: $a$ vs $0$). Two intersections.
Number of solutions $=\mathbf{2}$.
Correct Answer: 2