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Understanding how a function behaves at different points is a crucial part of calculus. Derivative analysis helps with this by showing us where the graph is increasing or decreasing.
In this case, for the function \( f(x) = \frac{2x^2}{x^2 - 1} \), we use the derivative to analyze the function's behavior. By applying the quotient rule, we find that the derivative of the function is \( f'(x) = \frac{-4x}{(x^2 - 1)^2} \).

The sign of this derivative tells us a lot:

• When \( f'(x) > 0 \), the function is increasing.
• When \( f'(x) < 0 \), the function is decreasing.

By examining intervals split by critical points and vertical asymptotes, we can ascertain whether the function increases or decreases within each interval. Here, a crucial observation is that the derivative changes sign at \( x = 0 \). This indicates a transition from increasing to decreasing behavior, suggesting a relative maximum at that point.

Relative extremes, such as maxima or minima, represent points where a function changes its increasing/decreasing behavior and reaches local high or low values.

For the function \( f(x) = \frac{2x^2}{x^2 - 1} \), after finding that the derivative \( f'(x) = \frac{-4x}{(x^2 - 1)^2} \) equals zero at \( x = 0 \), we detected a potential relative extremum. By examining \( f'(x) \), the sign changes from positive to negative as \( x \) moves through zero.

This shift implies a relative maximum at \( x = 0 \). The presence of a relative maximum indicates that \( x = 0 \) is a local high point in the graph of the function.

• No relative minima were found in this case based on our derivative analysis.
• Understanding these points helps in sketching the graph accurately.

Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a particular value. In rational functions like \( f(x) = \frac{2x^2}{x^2 - 1} \), these occur where the denominator equates to zero.

For this function, solving \( x^2 - 1 = 0 \) gives \( x = 1 \) and \( x = -1 \) as vertical asymptotes. As \( x \) nears these values, \( f(x) \) shoots off towards infinity, resulting in a dramatic spike or drop in the graph.

• Vertical asymptotes are particularly important as they dictate sections of the graph where the function is not defined.
• Identifying them is crucial for accurately sketching the function's behavior, particularly how it extends and bends at extreme points.

Remember, these asymptotes guide the graph's shape, narrowing predictions about its overall path.