The minimum value of (x-a)(x-b) is

Options

A. ab
B. `((a-b)^2)/4`
C. 0
D. `(-(a-b)^2)/4`

Answer

To find the minimum value of the expression \( (x - a)(x - b) \), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = (x - a)(x - b) \).

### Step 2: Expand the function
We can expand the function:
\[ f(x) = x^2 - (a + b)x + ab \]

### Step 3: Find the derivative
To find the minimum value, we first need to find the derivative of \( f(x) \):
\[ f'(x) = 2x - (a + b) \]

### Step 4: Set the derivative to zero
To find the critical points, set the derivative equal to zero:
\[ 2x - (a + b) = 0 \]
Solving for \( x \):
\[ 2x = a + b \implies x = \frac{a + b}{2} \]

### Step 5: Determine if it is a minimum or maximum
Next, we check the second derivative to determine whether this critical point is a minimum or maximum:
\[ f''(x) = 2 \]
Since \( f''(x) = 2 > 0 \), this indicates that the function has a minimum at \( x = \frac{a + b}{2} \).

### Step 6: Find the minimum value
Now we substitute \( x = \frac{a + b}{2} \) back into the function to find the minimum value:
\[ f\left(\frac{a + b}{2}\right) = \left(\frac{a + b}{2} - a\right)\left(\frac{a + b}{2} - b\right) \]
This simplifies to:
\[ = \left(\frac{b - a}{2}\right)\left(\frac{a - b}{2}\right) = \frac{(b - a)(a - b)}{4} = -\frac{(a - b)^2}{4} \]

### Conclusion
Thus, the minimum value of \( (x - a)(x - b) \) is:
\[ -\frac{(a - b)^2}{4} \]

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