To determine the exact values of inflection points for the function $f(x) = \frac{x}{e^{8x}}$, we need to find where the second derivative changes sign. Let's go through the steps:

Step 1: First Derivative

The function is: $f(x) = \frac{x}{e^{8x}}$ Using the quotient rule, where $u = x$ and $v = e^{8x}$: $f'(x) = \frac{v \cdot u' - u \cdot v'}{v^2} = \frac{e^{8x} \cdot 1 - x \cdot 8e^{8x}}{(e^{8x})^2}$ Simplifying this: $f'(x) = \frac{e^{8x} - 8x \cdot e^{8x}}{e^{16x}} = \frac{e^{8x}(1 - 8x)}{e^{16x}} = \frac{1 - 8x}{e^{8x}}$

Step 2: Second Derivative

Now, take the derivative of $f'(x) = \frac{1 - 8x}{e^{8x}}$ to find $f''(x)$.

Using the quotient rule again: $f''(x) = \frac{e^{8x} \cdot (-8) - (1 - 8x) \cdot 8e^{8x}}{(e^{8x})^2}$ Simplify this expression: $f''(x) = \frac{-8e^{8x} - 8(1 - 8x)e^{8x}}{e^{16x}} = \frac{-8e^{8x}(1 + 1 - 8x)}{e^{16x}} = \frac{-8(2 - 8x)}{e^{8x}}$