The problem asks to show that the function $f(x) = \frac{3 - 4x}{e^{2x}}$ has a minimum value when $x = \frac{5}{4}$.

Steps to verify this:

1. Function and Critical Points: To find the critical points, we first differentiate the function $f(x)$.

   The function is: $f(x) = \frac{3 - 4x}{e^{2x}}$

   Using the quotient rule for differentiation: $f'(x) = \frac{(e^{2x}) \cdot \frac{d}{dx}(3 - 4x) - (3 - 4x) \cdot \frac{d}{dx}(e^{2x})}{(e^{2x})^2}$ Simplifying: $f'(x) = \frac{-4e^{2x} - (3 - 4x) \cdot 2e^{2x}}{e^{4x}}$ $f'(x) = \frac{-4e^{2x} - 2(3 - 4x)e^{2x}}{e^{4x}}$ Factor out $e^{2x}$ from the numerator: $f'(x) = \frac{e^{2x}[-4 - 2(3 - 4x)]}{e^{4x}} = \frac{-4 - 6 + 8x}{e^{2x}} = \frac{8x - 10}{e^{2x}}$

2. Set the derivative equal to zero to find the critical points: $8x - 10 = 0$ $x = \frac{10}{8} = \frac{5}{4}$

   Therefore, the critical point is $x = \frac{5}{4}$.

3. Second Derivative Test: To confirm that this is a minimum, we can check the second derivative $f''(x)$. Differentiating $f'(x)$: $f'(x) = \frac{8x - 10}{e^{2x}}$ Differentiate again using the quotient rule: $f''(x) = \frac{8e^{2x} - (8x - 10) \cdot 2e^{2x}}{e^{4x}} = \frac{8e^{2x} - 2(8x - 10)e^{2x}}{e^{4x}}$ Simplifying: $f''(x) = \frac{e^{2x}[8 - 2(8x - 10)]}{e^{4x}} = \frac{8 - 16x + 20}{e^{2x}} = \frac{28 - 16x}{e^{2x}}$ At $x = \frac{5}{4}$: $f''\left(\frac{5}{4}\right) = \frac{28 - 16 \cdot \frac{5}{4}}{e^{2 \cdot \frac{5}{4}}} = \frac{28 - 20}{e^{5/2}} = \frac{8}{e^{5/2}} > 0$

   Since $f''\left(\frac{5}{4}\right) > 0$, the function has a local minimum at $x = \frac{5}{4}$.

Thus, the function $f(x)$ has a minimum value at $x = \frac{5}{4}$, which matches the condition given in the problem.

Would you like a more detailed explanation or any clarifications on this?

Here are 5 related questions you might find helpful:

1. How do we use the quotient rule in differentiation?
2. What does the second derivative test indicate about a function's concavity?
3. Can a function have more than one critical point?
4. How is the second derivative related to the shape of the graph?
5. What are other methods to find critical points of a function?

Tip: Always check the second derivative when you want to confirm whether a critical point is a minimum or maximum!