The function \( f(x) = x^2 + 1 \) is a quadratic function that opens upwards (since the coefficient of \( x^2 \) is positive). The minimum value of a quadratic function in the form \( ax^2 + bx + c \) occurs at the vertex, which can be found using the formula:
\[ x = -\frac{b}{2a} \]
For the function \( f(x) = x^2 + 1 \):
- \( a = 1 \)
- \( b = 0 \)
- \( c = 1 \)
Plugging in the values:
\[ x = -\frac{0}{2 \cdot 1} = 0 \]
Next, we find \( f(0) \):
\[ f(0) = (0)^2 + 1 = 1 \]
Therefore, the coordinate point at which the function has its minimum is:
\[ (0, 1) \]
So, the minimum occurs at the point \( (0, 1) \).
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