Understanding the Logarithmic Function
The problem asks us to find the minimum value of the function \(f(x) = \log_{10}(x^2 + 2x + 11)\).
The function \(f(x)\) is a logarithmic function with base 10. Since the base (10) is greater than 1, the logarithm is an increasing function. This means that the minimum value of \(f(x)\) will occur when the argument of the logarithm, which is the quadratic expression \(x^2 + 2x + 11\), reaches its minimum value.
Analyzing the Quadratic Argument
Let's focus on the quadratic expression inside the logarithm: \(g(x) = x^2 + 2x + 11\).
This is a quadratic function of the form \(ax^2 + bx + c\), where \(a=1\), \(b=2\), and \(c=11\). Since the coefficient \(a\) (which is 1) is positive, the parabola representing this quadratic function opens upwards, indicating that it has a minimum value.
Finding the Minimum Value of the Quadratic
The minimum value of a quadratic function \(ax^2 + bx + c\) occurs at its vertex. The x-coordinate of the vertex is given by the formula:
\(x = -\frac{b}{2a}\)
Substituting the values \(a=1\) and \(b=2\) from our quadratic expression:
\(x = -\frac{2}{2(1)} = -\frac{2}{2} = -1\)
Now, we find the minimum value of the quadratic expression by substituting \(x = -1\) back into \(g(x)\):
\(g(-1) = (-1)^2 + 2(-1) + 11\)
\(g(-1) = 1 - 2 + 11\)
\(g(-1) = 10\)
So, the minimum value of the quadratic expression \(x^2 + 2x + 11\) is 10.
Calculating the Minimum Function Value
Since the minimum value of the argument \(x^2 + 2x + 11\) is 10, we can now find the minimum value of the original function \(f(x) = \log_{10}(x^2 + 2x + 11)\):
\(f_{min} = \log_{10}(\text{minimum value of } x^2 + 2x + 11)\)
\(f_{min} = \log_{10}(10)\)
The logarithm of a number to the same base is always 1. Therefore:
\(f_{min} = 1\)
Thus, the minimum value of the function \(f(x) = \log_{10}(x^2 + 2x + 11)\) is 1.
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