Question:

Step 1: Identify the given function and its parameters. We are given $f(x) = (k - x^{10})^{1/10}$ where $k = 1025$. Step 2: Find the derivative of $f(x)$ using the chain rule. Applying the chain rule: $$f'(x) = \frac{1}{10}(k - x^{10})^{-9/10} \cdot (-10x^9) = \frac{-x^9}{(k - x^{10})^{9/10}}$$ Step 3: Calculate $f'(2)$. Substituting $x = 2$ and $k = 1025$: $$f'(2) = \frac{-(2)^9}{(1025 - 2^{10})^{9/10}} = \frac{-512}{(1025 - 1024)^{9/10}} = \frac{-512}{(1)^{9/10}} = -512$$ Step 4: Use the given condition to find $f'(a)$. We are given that $f'(2) = \frac{1}{f'(a)}$, so: $$-512 = \frac{1}{f'(a)}$$ Therefore: $$f'(a) = -\frac{1}{512}$$ Step 5: Set up the equation for $f'(a)$. Using the derivative formula: $$f'(a) = \frac{-a^9}{(k - a^{10})^{9/10}} = -\frac{1}{512}$$ Step 6: Recognize that $f$ is self-inverse and find the relationship between $a$ and $2$. For the function to satisfy the given condition, we need $f(a) = 2$. This means: $$(1025 - a^{10})^{1/10} = 2$$ Raising both sides to the 10th power: $$1025 - a^{10} = 2^{10} = 1024$$ Therefore: $$a^{10} = 1$$ Since $a \in \mathbb{N}$, we have $a = 1$. Step 7: Verify the solution by computing $f'(1)$. Substituting $a = 1$: $$f'(1) = \frac{-(1)^9}{(1025 - 1)^{9/10}} = \frac{-1}{(1024)^{9/10}} = \frac{-1}{(2^{10})^{9/10}} = \frac{-1}{2^9} = -\frac{1}{512}$$ Step 8: Confirm the condition is satisfied. We have: $$\frac{1}{f'(1)} = \frac{1}{-1/512} = -512 = f'(2)$$ ✓ Therefore, $a = \boxed{1}$, which corresponds to **Option 1**.