To determine whether the critical point \(x = -0.5\) corresponds to a minimum, maximum, or saddle point for the function \(f(x) = 1 + x + x^2\), we can use the second derivative test.
The second derivative of \(f(x)\) is the derivative of the first derivative:
\[f'(x) = 1 + 2x\]
Now, evaluate \(f''(-0.5)\):
\[f''(-0.5) = 2\]
The second derivative is positive, indicating that the function has a local minimum at \(x = -0.5\) (since the concavity changes from negative to positive).
Therefore, the correct answer is:
- Minima at \(x = -0.5\)