1. **Stating the problem:** We are given the function $$y = Ae^{2x} + Bxe^{2x}$$ and want to understand its shape and behavior. 2. **Formula and explanation:** This function is a combination of exponential growth $$e^{2x}$$ and a linear term $$x$$ multiplied by the same exponential. The constants $$A$$ and $$B$$ scale these parts. 3. **Factor the expression:** We can factor out $$e^{2x}$$: $$ y = e^{2x}(A + Bx) $$ 4. **Behavior analysis:** - The term $$e^{2x}$$ grows very fast as $$x$$ increases. - The term $$(A + Bx)$$ is linear, so it can change sign depending on $$x$$. 5. **Critical points (extrema):** To find extrema, differentiate: $$ y' = \frac{d}{dx}[e^{2x}(A + Bx)] = e^{2x}(2)(A + Bx) + e^{2x}(B) = e^{2x}(2A + 2Bx + B) $$ Set derivative to zero: $$ 0 = e^{2x}(2A + B + 2Bx) $$ Since $$e^{2x} \neq 0$$, solve: $$ 2A + B + 2Bx = 0 \implies 2Bx = -2A - B \implies x = \frac{-2A - B}{2B} $$ 6. **Intercepts:** - At $$x=0$$: $$ y = A e^{0} + B \cdot 0 \cdot e^{0} = A $$ - At $$y=0$$: $$ 0 = e^{2x}(A + Bx) \implies A + Bx = 0 \implies x = -\frac{A}{B} $$ 7. **Summary:** - The function grows exponentially. - It has one critical point at $$x = \frac{-2A - B}{2B}$$. - It crosses the x-axis at $$x = -\frac{A}{B}$$. This explains the shape of the graph with exponential growth modulated by a linear term.