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To determine the concavity of a function, one robust method is using the second derivative test. Concavity refers to the way a curve bends, telling us whether it is curved upwards (concave up) or downwards (concave down). If a function's second derivative denoted as \( f''(x) \) is negative over an interval, the graph of the function is concave down on that interval. Conversely, if \( f''(x) \) is positive, the function is considered concave up.
In our problem, the function \( g(x) = \frac{1}{f(x)} \) requires showing concavity through its second derivative. The given condition \( f''(x) > 0 \) suggests that \( f(x) \) itself is concave up. However, to understand the concavity of \( g(x) \), we rely on calculating \( g''(x) \). When analyzing \( g''(x) \), we derive concavity information by comparing the effects of its components' behavior. If the result of \( g''(x) \) is negative, it confirms that \( g(x) \) is concave down everywhere.

The chain rule is a fundamental concept in calculus used to differentiate composite functions. When a function is composed of two or more functions, the chain rule simplifies the process of finding its derivative.
In our example, we have \( g(x) = \frac{1}{f(x)} \), a composite function, since it involves \( f(x) \) in the denominator. To differentiate such a function, we apply the chain rule combined with the quotient rule. This shows us that \( g'(x) = -\frac{f'(x)}{(f(x))^2} \). The understanding here is that the derivative of \( 1/g(x) \), which has \( f(x) \) as its inner function, directly leads us to use the chain rule.
By doing so, the intricate relation between \( f(x) \) and \( g(x) \) becomes unravelled efficiently, allowing a foothold to further explore the second derivative and assess concavity. Without the chain rule, such derivative computations would be cumbersome and significantly more complex.

Function behavior describes how a function acts over different intervals, giving insight into trends such as increasing, decreasing, and concavity. In the given exercise, we examine the behavior of \( f(x) \) and consequently, \( g(x) \).
For \( f(x) \), which is negative everywhere and has a positive second derivative, we understand that \( f(x) \) tends to curve upwards despite being negative, implying an upward concaving trend. Meanwhile, \( g(x) \) \( = \frac{1}{f(x)} \) is dependent on \( f(x) \).
Because \( f(x) \) is negative, \( g(x) \)'s positivity or negativity is the inverse of \( f(x) \). Ultimately, when \( g''(x) \) is calculated, its negativity reveals \( g(x) \) as consistently concave down—indicating the opposite bending of the curve compared to \( f(x) \). This clarity in understanding the functional behavior assists in predicting and visually identifying key traits.