A quadratic function f(x) satisfies `f(x)ge0` for all real x. If f(2)=0 and f(4)=12, then the value of f(6) is :

Options

A. 12
B. 24
C. 48
D. 64

Explanation

To solve the problem, we need to find the value of the quadratic function \( f(x) \) at \( x = 6 \), given that \( f(x) \geq 0 \) for all real \( x \), \( f(2) = 0 \), and \( f(4) = 12 \).

### Step-by-Step Solution:

1. **Understanding the Quadratic Function**:
Since \( f(x) \geq 0 \) for all real \( x \) and \( f(2) = 0 \), we can express \( f(x) \) in the form:
\[ f(x) = k(x - 2)^2 \]
where \( k \) is a positive constant. This form ensures that the function is always non-negative and has a root at \( x = 2 \).

**Hint**: Remember that a quadratic function that is always non-negative can be expressed as a square of a binomial multiplied by a positive constant.

2. **Using the Given Information**:
We know that \( f(4) = 12 \). We can substitute \( x = 4 \) into our function:
\[ f(4) = k(4 - 2)^2 = k(2)^2 = 4k \]
Setting this equal to 12 gives us:
\[ 4k = 12 \]

**Hint**: Substitute the known value into the function to find the constant \( k \).

3. **Solving for \( k \)**:
To find \( k \), divide both sides of the equation by 4:
\[ k = \frac{12}{4} = 3 \]

**Hint**: Isolate the variable to find its value.

4. **Writing the Function**:
Now that we have \( k \), we can write the complete function:
\[ f(x) = 3(x - 2)^2 \]

**Hint**: Substitute the value of \( k \) back into the function to express it fully.

5. **Finding \( f(6) \)**:
Now we need to find \( f(6) \):
\[ f(6) = 3(6 - 2)^2 = 3(4)^2 = 3 \times 16 = 48 \]

**Hint**: Substitute \( x = 6 \) into the function to find the desired value.

6. **Final Answer**:
Therefore, the value of \( f(6) \) is:
\[ \boxed{48} \]