Let f (x) = [x], where [x] denotes the greatest integer less than or equal to x . If a `= sqrt(2011^(2) + 2012)`, then the value of f (x) is equal to

Options

A. 2010
B. 2011
C. 2012
D. 2013

Answer

To solve the problem, we need to find the value of \( f(a) \) where \( a = \sqrt{2011^2 + 2012} \) and \( f(x) = [x] \) denotes the greatest integer less than or equal to \( x \).

### Step 1: Calculate \( a \)
First, we need to compute \( a \):
\[ a = \sqrt{2011^2 + 2012} \]

### Step 2: Calculate \( 2011^2 \)
Now, let's calculate \( 2011^2 \):
\[ 2011^2 = 4044121 \]

### Step 3: Add \( 2012 \) to \( 2011^2 \)
Next, we add \( 2012 \) to \( 2011^2 \):
\[ 4044121 + 2012 = 4046133 \]

### Step 4: Take the square root
Now, we take the square root of the sum:
\[ a = \sqrt{4046133} \]

### Step 5: Estimate \( \sqrt{4046133} \)
To find \( \sqrt{4046133} \), we can estimate it. We know:
\[ 2011^2 = 4044121 \quad \text{and} \quad 2012^2 = 4048144 \]
Since \( 4044121 < 4046133 < 4048144 \), we can conclude:
\[ 2011 < \sqrt{4046133} < 2012 \]

### Step 6: Find the greatest integer less than or equal to \( a \)
Since \( a \) is between \( 2011 \) and \( 2012 \), the greatest integer less than or equal to \( a \) is:
\[ f(a) = [a] = 2011 \]

### Final Answer
Thus, the value of \( f(a) \) is:
\[ \boxed{2011} \]