If `[x]` is the greatest integer less than or equal to `x` and `(x)` be the least integer greater than or equal to `x` and `[x]^(2)+(x)^(2)gt25` then `x` belongs to

Options

A. (a) `[3,4]`
B. (b) `(-oo,-4]`
C. (c) `[4,oo)`
D. (d) `(-oo,-4]uu[4,oo)`

Answer

To solve the problem, we need to analyze the inequality given by the expression involving the greatest integer function \([x]\) and the least integer function \((x)\).

### Step 1: Understand the Functions
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The least integer function \((x)\) gives the smallest integer greater than or equal to \(x\).

### Step 2: Set Up the Inequality
We are given the inequality:
\[ [x]^2 + (x)^2 > 25 \]

### Step 3: Case 1 - When \(x\) is an Integer
If \(x\) is an integer, then \([x] = (x) = x\). Therefore, the inequality simplifies to:
\[ x^2 + x^2 > 25 \]
This can be rewritten as:
\[ 2x^2 > 25 \]
Dividing both sides by 2:
\[ x^2 > \frac{25}{2} \]
Taking the square root:
\[ |x| > \frac{5}{\sqrt{2}} \approx 3.54 \]
Thus, the integer solutions for \(x\) are:
\[ x \leq -4 \quad \text{or} \quad x \geq 4 \]

### Step 4: Case 2 - When \(x\) is Not an Integer
If \(x\) is not an integer, then we can express \(x\) as:
\[ x = n + d \quad \text{where } n = [x] \text{ (an integer) and } 0 < d < 1 \]
In this case, \([x] = n\) and \((x) = n + 1\). The inequality becomes:
\[ n^2 + (n + 1)^2 > 25 \]
Expanding this:
\[ n^2 + (n^2 + 2n + 1) > 25 \]
This simplifies to:
\[ 2n^2 + 2n + 1 > 25 \]
Rearranging gives:
\[ 2n^2 + 2n - 24 > 0 \]
Dividing by 2:
\[ n^2 + n - 12 > 0 \]
Factoring:
\[ (n + 4)(n - 3) > 0 \]

### Step 5: Analyze the Quadratic Inequality
To solve the inequality \((n + 4)(n - 3) > 0\), we find the critical points \(n = -4\) and \(n = 3\). Testing intervals:
- For \(n < -4\), both factors are negative, so the product is positive.
- For \(-4 < n < 3\), one factor is negative and one is positive, so the product is negative.
- For \(n > 3\), both factors are positive, so the product is positive.

Thus, the solution for \(n\) is:
\[ n < -4 \quad \text{or} \quad n > 3 \]

### Step 6: Determine Possible Values of \(x\)
- If \(n < -4\), then \(n\) can be \(-5, -6, -7, \ldots\) which gives \(x < -4\).
- If \(n > 3\), then \(n\) can be \(4, 5, 6, \ldots\) which gives \(x \geq 4\).

### Final Conclusion
Combining both cases, we conclude that \(x\) belongs to the intervals:
\[ (-\infty, -4) \cup [4, \infty) \]