Find the minimum value of `3x^(2) - 6x + 11`

Answer

To find the minimum value of the quadratic function \( f(x) = 3x^2 - 6x + 11 \), we can follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function with respect to \( x \).

\[ f'(x) = \frac{d}{dx}(3x^2) - \frac{d}{dx}(6x) + \frac{d}{dx}(11) \]

### Step 2: Apply the power rule
Using the power rule of differentiation, we calculate:

- The derivative of \( 3x^2 \) is \( 6x \).
- The derivative of \( 6x \) is \( 6 \).
- The derivative of a constant \( 11 \) is \( 0 \).

So, we have:

\[ f'(x) = 6x - 6 \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:

\[ 6x - 6 = 0 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):

\[ 6x = 6 \implies x = 1 \]

### Step 5: Find the minimum value
Next, we substitute \( x = 1 \) back into the original function to find the minimum value:

\[ f(1) = 3(1)^2 - 6(1) + 11 \]

Calculating this gives:

\[ f(1) = 3 - 6 + 11 = 8 \]

### Conclusion
Thus, the minimum value of the function \( 3x^2 - 6x + 11 \) is \( 8 \).

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