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`

Answer

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: Differentiate the function
To find the critical points, we differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = 2x - 8 \]

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

### Step 4: Determine if it is a minimum or maximum
Next, we find the second derivative to determine the nature of the critical point:
\[ f''(x) = 2 \]
Since \( f''(x) = 2 > 0 \), this indicates that the function has a local minimum at \( x = 4 \).

### 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:
\[ f(4) = 16 - 32 + 17 = 1 \]

Thus, the minimum value of the expression \( x^2 - 8x + 17 \) is **1**.

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