Evaluate Statements on Inverse Function Fixed Points and Shifted Equations

To evaluate the given statements, we first analyze the function $f : [1, \infty) \rightarrow [1, \infty)$ defined by: $f(x) = (x - 1)^4 + 1$

Since $f'(x) = 4(x - 1)^3 \ge 0$ for all $x \in [1, \infty)$, the function $f(x)$ is strictly increasing on $[1, \infty)$.

To find the inverse function $f^{-1}(x)$, we set $y = (x - 1)^4 + 1$: $(x - 1)^4 = y - 1 \implies x - 1 = (y - 1)^{1/4} \implies x = (y - 1)^{1/4} + 1$ Thus, the inverse function $f^{-1} : [1, \infty) \rightarrow [1, \infty)$ is given by: $f^{-1}(x) = (x - 1)^{1/4} + 1$

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Analysis of Statement (I):

The set $S = \{x \in [1, \infty) : f(x) = f^{-1}(x)\}$.

Since $f(x)$ is a strictly increasing function on $[1, \infty)$, the solutions to the equation $f(x) = f^{-1}(x)$ are precisely the fixed points of $f(x)$, i.e., the solutions to $f(x) = x$.

Setting $f(x) = x$: $(x - 1)^4 + 1 = x$ $(x - 1)^4 - (x - 1) = 0$ $(x - 1)\left[(x - 1)^3 - 1\right] = 0$

This gives two real solutions:

1. $x - 1 = 0 \implies x = 1$
2. $(x - 1)^3 - 1 = 0 \implies (x - 1)^3 = 1 \implies x - 1 = 1 \implies x = 2$

Both $x = 1$ and $x = 2$ belong to the domain $[1, \infty)$. Therefore, the set $S = \{1, 2\}$ contains exactly two elements.

Hence, Statement (I) is TRUE.

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Analysis of Statement (II):

The set $S = \{x \in [1, \infty) : f(x) = f^{-1}(x + 1)\}$.

Using $f(x) = (x - 1)^4 + 1$ and $f^{-1}(x + 1) = ((x + 1) - 1)^{1/4} + 1 = x^{1/4} + 1$: $(x - 1)^4 + 1 = x^{1/4} + 1$ $(x - 1)^4 = x^{1/4}$

Consider the continuous function $g(x) = (x - 1)^4 - x^{1/4}$ for $x \in [1, \infty)$:

• At $x = 2$: $g(2) = (2 - 1)^4 - 2^{1/4} = 1 - 2^{1/4} \approx 1 - 1.189 = -0.189 < 0$
• At $x = 2.2$: $g(2.2) = (1.2)^4 - (2.2)^{1/4} = 2.0736 - 1.218 = 0.8556 > 0$

By the Intermediate Value Theorem, since $g(x)$ is continuous and changes sign on the interval $(2, 2.2)$, there exists at least one real root $x_0 \in (2, 2.2)$ such that $g(x_0) = 0$, which means $f(x_0) = f^{-1}(x_0 + 1)$.

Thus, the set $S$ is non-empty.

Hence, Statement (II) is FALSE.

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Conclusion:

Only Statement (I) is TRUE, which corresponds to Option A.