| Describe in detail what you need | The graph of f(x) = x2 – 2x + 1 has a local minimum value at x = -1. True or False |
| How many pages | 1 |
To determine whether the statement "The graph of f(x) = x² - 2x + 1 has a local minimum value at x = -1" is true or false, we will analyze the function step by step.
The function we are examining is given by:
[ f(x) = x^2 - 2x + 1 ]
This is a quadratic function, which can generally be expressed in the standard form:
[ f(x) = ax^2 + bx + c ]
In our case:
The graph of any quadratic function is a parabola. Since the coefficient of (x^2) (which is (a)) is positive, we know that this parabola opens upwards, indicating that it may have a local minimum.
The vertex of a quadratic function provides the location of its minimum or maximum. The x-coordinate of the vertex can be calculated using the formula:
[ x = -\frac{b}{2a} ]
Substituting the values of (b) and (a) into the formula, we get:
[ x = -\frac{-2}{2 \times 1} = \frac{2}{2} = 1 ]
The vertex of the function (f(x) = x^2 - 2x + 1) is therefore located at (x = 1).
Now, let's find out the value of the function at the vertex (x = 1):
[ f(1) = (1)^2 - 2(1) + 1 = 1 - 2 + 1 = 0 ]
Thus, the coordinate of the vertex is ((1, 0)), and it is indeed a local minimum value.
Since the vertex, which corresponds to the local minimum value of the function, is located at (x = 1) and not at (x = -1), we can conclude that the original statement is False.
The function (f(x) = x^2 - 2x + 1) does not have a local minimum at (x = -1) but rather at (x = 1). Therefore, the correct interpretation of the local minimum of this quadratic function shows the importance of analyzing the vertex correctly.
In summary, the verification process of the function's characteristics confirms that the statement provided is indeed false, as it does not align with the mathematical principles governing quadratic functions.