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The problem you're working on involves a quadratic function \( f(x) = ax^2 + bx + c \) with certain conditions given. Let's break it down step by step:

The equation provided is:

\[
f^2(-1) + f^2(0) + f^2(1) + 30 = 4f(0) + 10f(1) + 2f(-1)
\]

From this, we need to:

1. Find the coefficients \( a \), \( b \), and \( c \) of the quadratic function \( f(x) = ax^2 + bx + c \).
2. Once we determine the function \( f(x) \), we will evaluate the different ranges of inverse trigonometric functions like \( \sin^{-1}(f(x)) \), \( \cos^{-1}(f(x)) \), etc., based on the given answer choices.

### Step 1: Evaluate \( f(-1), f(0), f(1) \)

Using the general form of \( f(x) = ax^2 + bx + c \):

- \( f(-1) = a(-1)^2 + b(-1) + c = a - b + c \)
- \( f(0) = a(0)^2 + b(0) + c = c \)
- \( f(1) = a(1)^2 + b(1) + c = a + b + c \)

Substituting these into the equation:

\[
(a - b + c)^2 + c^2 + (a + b + c)^2 + 30 = 4c + 10(a + b + c) + 2(a - b + c)
\]

Now, solving this equation will help us find the values of \( a \), \( b \), and \( c \).

Let me calculate the solution for this system.The system provides two possible solutions for \(a\), depending on the values of \(b\) and \(c\). The solutions for \(a\) are:

1. \( a = -\frac{c + \sqrt{-4b^2 + 16b - 2c^2 + 8c - 24}}{2} + 3 \)
2. \( a = -\frac{c - \sqrt{-4b^2 + 16b - 2c^2 + 8c - 24}}{2} + 3 \)

Both are functions of \(b\) and \(c\). Since the problem provides a constraint, we need to consider how these interact with the inverse trigonometric ranges to determine the most appropriate form of \(f(x)\) and then analyze the range of the corresponding trigonometric inverse functions.

Would you like to proceed with the range analysis or focus on solving for specific values of \(a\), \(b\), and \(c\)? 

Here are 5 related questions that could help expand on this:

1. What are the ranges of inverse trigonometric functions like \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \)?
2. How can the discriminant of the quadratic expression help in simplifying the values for \(a\), \(b\), and \(c\)?
3. What happens to the quadratic equation when certain values of \(b\) and \(c\) are chosen?
4. How can we use the given answer options (A to D) to further refine the solution for \(f(x)\)?
5. What is the significance of \( f(x) \) being inside inverse trigonometric functions, and how does this affect their range?

**Tip:** The range of a quadratic function depends on the sign of its leading coefficient \(a\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.

If f(x) = ax^2 + bx + c is such that f^2(-1) + f^2(0) + f^2(1) + 30 = 4f(0) + 10f(1) + 2f(-1), then find the range of inverse trigonometric functions of f(x).

This problem involves finding the range of inverse trigonometric functions such as sin⁻¹(f(x)), cos⁻¹(f(x)), tan⁻¹(f(x)), and cot⁻¹(f(x)) given a quadratic function f(x) = ax^2 + bx + c with specific conditions. Detailed solution provided for high school students.

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