1+(х-1)^2,1<=х<=2

To analyze the expression \( f(x) = 1 + (x - 1)^2 \) over the interval \( 1 \leq x \leq 2 \), let's proceed with the following steps:

1. Determine the function: \[ f(x) = 1 + (x - 1)^2 \]

2. Identify the behavior of the function over the interval:

   • At \( x = 1 \): \[ f(1) = 1 + (1 - 1)^2 = 1 + 0 = 1 \]
   • At \( x = 2 \): \[ f(2) = 1 + (2 - 1)^2 = 1 + 1 = 2 \]

3. Analyze the function: The term \((x - 1)^2\) is always non-negative and achieves its minimum value of \(0\) at \(x = 1\). Hence, \(f(x)\) is increasing on the interval \([1, 2]\).

4. Conclude the behavior:

   • The minimum value of \( f(x) \) on the interval \( [1, 2] \) is \( 1 \) (at \( x = 1 \)).
   • The maximum value of \( f(x) \) on the interval \( [1, 2] \) is \( 2 \) (at \( x = 2 \)).

In summary, \( f(x) = 1 + (x - 1)^2 \) takes values between \( 1 \) and \( 2 \) in the interval \( [1, 2] \).