1. **State the problem:** We are given the function $$f(x) = (2x - 5)e^x$$ and need to find the exact coordinates of the minimum turning point A. 2. **Find the derivative:** To find turning points, we differentiate $$f(x)$$ using the product rule: $$f'(x) = \frac{d}{dx}[(2x - 5)e^x] = (2x - 5)\frac{d}{dx}[e^x] + e^x \frac{d}{dx}[2x - 5] = (2x - 5)e^x + 2e^x = e^x(2x - 5 + 2) = e^x(2x - 3)$$ 3. **Set derivative to zero:** Turning points occur where $$f'(x) = 0$$: $$e^x(2x - 3) = 0$$ Since $$e^x \neq 0$$ for all real $$x$$, we solve: $$2x - 3 = 0 \implies x = \frac{3}{2}$$ 4. **Find y-coordinate:** Substitute $$x = \frac{3}{2}$$ into $$f(x)$$: $$f\left(\frac{3}{2}\right) = \left(2 \times \frac{3}{2} - 5\right) e^{\frac{3}{2}} = (3 - 5) e^{1.5} = -2 e^{1.5}$$ 5. **Coordinates of A:** The minimum turning point A is at: $$\left(\frac{3}{2}, -2 e^{\frac{3}{2}}\right)$$ 6. **Confirm minimum:** Since $$f'(x)$$ changes from negative to positive at $$x=\frac{3}{2}$$ (or by second derivative test), A is a minimum. **Final answer:** $$A = \left(\frac{3}{2}, -2 e^{\frac{3}{2}}\right)$$