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The Extreme Value Theorem is a fundamental concept in calculus that addresses the existence of extreme values in continuous functions. This theorem states that if a function is continuous on a closed interval \([a, b]\), then the function must attain both a maximum and a minimum value within that interval. The maximum or minimum can occur either at the critical points or at the endpoints of the interval.
The theorem is particularly useful in scenarios like the given exercise where we need to find the absolute extreme values of a function within a particular range. We apply this theorem to ensure that, when evaluating a function over a closed interval, we have thoroughly checked both critical points and endpoints to ascertain the extreme values. Without this guarantee of completeness from the Extreme Value Theorem, some function evaluations might be overlooked, leading to incorrect conclusions.

Critical points of a function are points where its derivative is either zero or undefined. To identify these points, we take the derivative of the given function and solve for points where this derivative equals zero or does not exist.
For the function \( f(x) = \frac{1}{x^2 + 1} \), the derivative \( f'(x) = \frac{-2x}{(x^2 + 1)^2} \) was calculated using the quotient rule. We set the derivative equal to zero to find the critical points:

• \(-2x = 0\) implies \(x = 0\), which is our critical point in the interval \([-3, 3]\).

Critical points are significant because they provide potential locations for local maxima or minima, where changes in the slope of a curve suggest transitions between increasing and decreasing behavior. It is crucial to confirm whether these are indeed maxima or minima by evaluating the function at these points.

The derivative of a function provides the slope of the tangent line to the graph of the function at any given point. It is a measure of how the function's output changes in response to changes in its input.
In our exercise, the derivative \( f'(x) = \frac{-2x}{(x^2 + 1)^2} \) was obtained using the quotient rule, highlighting how changes in \( x \) affect the shape of the graph \( f(x) \).
Derivatives are critical in finding critical points because they show where the function's rate of change halts or shifts direction. Understanding the concept of derivatives helps to determine where these shifts occur and whether they result in maxima, minima, or neither. This is key in analyzing the behavior of functions within specified intervals, as in the original problem.

The quotient rule is a technique in calculus for differentiating functions that are expressed as the ratio of two differentiable functions. The rule states that if you have a function \( f(x) = \frac{g(x)}{h(x)} \), then the derivative \( f'(x) \) is given by:

• \( f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2} \)

In our solution, the function \( f(x) = \frac{1}{x^2 + 1} \) was differentiated using the quotient rule.
Here, \( g(x) = 1 \) and \( h(x) = x^2 + 1 \). Plugging these into the quotient rule formula yielded the derivative \( f'(x) = \frac{-2x}{(x^2 + 1)^2} \).
This rule is essential for handling more complex functions where a straightforward application of basic derivative rules isn't feasible. It ensures that we can accurately assess the derivative of such functions, which is critical for finding critical points and understanding the function's behavior over specific intervals.