Let f be a function with f(x)>0 for all x . Set g=1 / f (a)...

Let $f$ be a function with $f(x)>0$ for all $x .$ Set $g=1 / f$
(a) If $f$ is increasing in an interval around $x_{0},$ what about $g ?$
(b) If $f$ has a local maximum at $x_{1},$ what about $g$ ?
(c) If $f$ is concave down at $x_{2},$ what about $g$ ?

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Key Concepts

Monotonicity

Monotonicity refers to the property of a function to be either entirely non-increasing or non-decreasing over an interval. In the context of a reciprocal function, understanding how monotonicity behaves under transformation is important because a strictly increasing positive function will yield a strictly decreasing reciprocal, reflecting the inverse relationship.

Local Extrema

Local extrema are points where a function reaches a local minimum or maximum relative to nearby points. When applying a reciprocal transformation to a function with a local maximum, the nature of these extrema can change; specifically, if the original function attains a local maximum, its reciprocal is likely to attain a local minimum, because the highest value of the original function produces the smallest value in the reciprocal.

Concavity

Concavity describes how a function curves, indicating whether it bends upward or downward. Examining the concavity of a function is critical when considering its transformed behavior. For a reciprocal function, the transformation can alter the curvature properties; for instance, a function that is concave down may not directly translate to the reciprocal function having the same concavity. Analyzing the second derivative is essential to understand this change in curvature.

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