The domain of the function `f(x)=sqrt(10-sqrt(x^4-21 x^2))` is
a) `[5,oo)`
b. `[-sqrt(21),sqrt(21)]`
c. `[-5,-sqrt(21)]uu[sqrt(21),5]uu{0]`
d. `(-oo,-5)`

Options

A. `[5,oo]`
B. `[-sqrt(21),sqrt(21)]`
C. `[-5-sqrt(21]]uu[sqrt(21),5)]uu{0}`
D. `(-oo,-5)`

Explanation

To find the domain of the function \( f(x) = \sqrt{10 - \sqrt{x^4 - 21x^2}} \), we need to ensure that the expression inside the square roots is non-negative. This leads us to two conditions:

1. The expression under the outer square root must be non-negative:
\[ 10 - \sqrt{x^4 - 21x^2} \geq 0 \]
This implies:
\[ \sqrt{x^4 - 21x^2} \leq 10 \]

2. The expression under the inner square root must also be non-negative:
\[ x^4 - 21x^2 \geq 0 \]

### Step 1: Solve the inner condition

Starting with the inner condition:
\[ x^4 - 21x^2 \geq 0 \]
We can factor this expression:
\[ x^2(x^2 - 21) \geq 0 \]
This gives us two factors:
- \( x^2 \geq 0 \) (which is always true since squares are non-negative)
- \( x^2 - 21 \geq 0 \) which simplifies to:
\[ x^2 \geq 21 \]
Thus, we have:
\[ x \leq -\sqrt{21} \quad \text{or} \quad x \geq \sqrt{21} \]

### Step 2: Solve the outer condition

Now, we consider the outer condition:
\[ \sqrt{x^4 - 21x^2} \leq 10 \]
Squaring both sides gives:
\[ x^4 - 21x^2 \leq 100 \]
Rearranging this, we have:
\[ x^4 - 21x^2 - 100 \leq 0 \]
Let \( y = x^2 \). Then the inequality becomes:
\[ y^2 - 21y - 100 \leq 0 \]
We can solve this quadratic inequality using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{21 \pm \sqrt{(-21)^2 - 4 \cdot 1 \cdot (-100)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 441 + 400 = 841 \quad \Rightarrow \quad \sqrt{841} = 29 \]
Thus, the roots are:
\[ y = \frac{21 \pm 29}{2} \]
Calculating the two roots:
1. \( y_1 = \frac{50}{2} = 25 \)
2. \( y_2 = \frac{-8}{2} = -4 \) (not relevant since \( y \) must be non-negative)

So, we have:
\[ y^2 - 21y - 100 \leq 0 \quad \Rightarrow \quad 0 \leq y \leq 25 \]
Substituting back \( y = x^2 \):
\[ 0 \leq x^2 \leq 25 \quad \Rightarrow \quad -5 \leq x \leq 5 \]

### Step 3: Combine the conditions

Now we combine the results from the inner and outer conditions:
1. From the inner condition: \( x \leq -\sqrt{21} \) or \( x \geq \sqrt{21} \)
2. From the outer condition: \( -5 \leq x \leq 5 \)

The valid intervals are:
- From \( -5 \) to \( -\sqrt{21} \)
- From \( \sqrt{21} \) to \( 5 \)

Thus, the domain of the function \( f(x) \) is:
\[ [-5, -\sqrt{21}] \cup [\sqrt{21}, 5] \]

### Final Answer
The correct option is:
c) \( [-5, -\sqrt{21}] \cup [\sqrt{21}, 5] \)