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To understand the behavior of a function, derivatives play a crucial role. A derivative tells us how a function changes at any given point, which is pivotal for determining the slope of the graph. In simpler terms, it shows the rate at which one quantity changes with respect to another. For the quadratic function provided, \( f(x) = x^2 - x + 1 \), its derivative is \( f'(x) = 2x - 1 \). This derivative is a linear function of \( x \), indicating that the slope of the original function changes linearly as \( x \) changes.

When you set a derivative equal to zero, like \( 2x - 1 = 0 \), you're finding the points where the slope of the graph is flat, meaning it could be a maximum, minimum, or neither (but for a quadratic, it typically tells us about a minimum or maximum). This process helps identify the extrema, or critical points, of the function.

Extrema are critical points where a function takes on a maximum or minimum value. Identifying these points is especially important in analyzing quadratic functions. For the function \( f(x) = x^2 - x + 1 \), we calculate its derivative \( f'(x) = 2x - 1 \) and set the equation \( 2x - 1 = 0 \) to solve for \( x \). This yields \( x = 0.5 \).

After finding the \( x \)-coordinate of an extremum, substituting it back into the original function gives us the \( y \)-value or the function's extrema. Thus, \( f(0.5) = 0.5^2 - 0.5 + 1 = 0.75 \). Since the parabola opens upwards (due to the positive coefficient of the \( x^2 \) term), this point \( (0.5, 0.75) \) is a minimum. Such calculations allow us to determine that the lowest value reached by this function is 0.75, meaning the function never goes negative.

Function analysis involves understanding the function's overall behavior over its domain. By examining the given quadratic function \( f(x) = x^2 - x + 1 \), we can predict how the function behaves at various points.

• The parabolic shape ensures that as \( x \) becomes very large or very small, the value of \( f(x) \) increases dramatically. This is due to the \( x^2 \) term, which grows faster than the linear terms.
• The function's minimum value, which we've identified at \( (0.5, 0.75) \), confirms that the function stays above 0 everywhere on its graph. Therefore, it does not have any negative values.
• Moreover, knowing that the graph never intersects the x-axis assures us that \( f(x) \) is always positive.

Such analysis provides a complete picture of how a quadratic function behaves across its domain and for any values of \( x \). This understanding can help predict outcomes and make informed conclusions about the function's characteristics.