The function `f(x)=(x-1)^(2)` has a minimum at `x` is equal to

Options

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

Explanation

To find the minimum of the function \( f(x) = (x - 1)^2 \), we will follow these steps:

### Step 1: Find the first derivative of the function
The first step is to differentiate the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}[(x - 1)^2] \]

Using the power rule:

\[ f'(x) = 2(x - 1) \]

### Step 2: Set the first derivative equal to zero
To find the critical points, we set the first derivative equal to zero:

\[ f'(x) = 0 \implies 2(x - 1) = 0 \]

### Step 3: Solve for \( x \)
Now, we solve the equation:

\[ 2(x - 1) = 0 \implies x - 1 = 0 \implies x = 1 \]

### Step 4: Find the second derivative
Next, we need to find the second derivative to determine whether this critical point is a minimum or maximum.

\[ f''(x) = \frac{d}{dx}[2(x - 1)] = 2 \]

### Step 5: Evaluate the second derivative at the critical point
Now, we evaluate the second derivative at \( x = 1 \):

\[ f''(1) = 2 \]

Since \( f''(1) > 0 \), this indicates that the function has a local minimum at \( x = 1 \).

### Conclusion
Thus, the function \( f(x) = (x - 1)^2 \) has a minimum at:

\[ \boxed{1} \]

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