If `x` is real, then the minimum value of the expression `x^2-8x+17` is

Options

A. `-1`
B. 0
C. 1
D. 2

Explanation

To find the minimum value of the expression \( x^2 - 8x + 17 \), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x^2 - 8x + 17 \).

### Step 2: Find the first derivative
To find the critical points, we first need to differentiate the function with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(x^2 - 8x + 17) \]
Using the power rule of differentiation, we get:
\[ f'(x) = 2x - 8 \]

### Step 3: Set the first derivative to zero
To find the critical points, we set the first derivative equal to zero:
\[ 2x - 8 = 0 \]
Solving for \( x \):
\[ 2x = 8 \\ x = \frac{8}{2} = 4 \]

### Step 4: Verify if it is a minimum using the second derivative
Now, we need to check if this critical point is a minimum by finding the second derivative:
\[ f''(x) = \frac{d}{dx}(2x - 8) \]
Calculating the second derivative, we find:
\[ f''(x) = 2 \]
Since \( f''(x) = 2 > 0 \), this indicates that the function is concave up at \( x = 4 \), confirming that it is indeed a minimum point.

### Step 5: Calculate the minimum value
Now, we substitute \( x = 4 \) back into the original function to find the minimum value:
\[ f(4) = 4^2 - 8 \cdot 4 + 17 \]
Calculating this:
\[ f(4) = 16 - 32 + 17 = 1 \]

### Conclusion
The minimum value of the expression \( x^2 - 8x + 17 \) is \( \boxed{1} \).

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