It y=g(x) is a curve which is obtained by the reflecton of y=f(x)=(e^x-e^-x)/(2) by the line y=x

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It y=g(x) is a curve which is obtained by the reflecton of...

It $\mathrm{y}=\mathrm{g}(\mathrm{x})$ is a curve which is obtained by the reflecton of $y=f(x)=\frac{e^{x}-e^{-x}}{2}$ by the line $y=x$ then (A) $\mathrm{y}=\mathrm{g}(\mathrm{x})$ has exactly one tangent parallel to $\mathrm{x}$-axis (B) $\mathrm{y}=\mathrm{g}(\mathrm{x})$ has no tangent parallel to $\mathrm{y}$-axis (C) The tangent to $y=g(x)$ at $(0,0)$ is $y=x$ (D) $g(x)$ has no extremum

It $\mathrm{y}=\mathrm{g}(\mathrm{x})$ is a curve which is obtained by the reflecton of $y=f(x)=\frac{e^{x}-e^{-x}}{2}$ by the line $y=x$ then
(A) $\mathrm{y}=\mathrm{g}(\mathrm{x})$ has exactly one tangent parallel to $\mathrm{x}$-axis
(B) $\mathrm{y}=\mathrm{g}(\mathrm{x})$ has no tangent parallel to $\mathrm{y}$-axis
(C) The tangent to $y=g(x)$ at $(0,0)$ is $y=x$
(D) $g(x)$ has no extremum

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Key Concepts

Monotonicity and Extremum Properties

A monotonic function is one that is either entirely non-increasing or non-decreasing, which implies the absence of local extrema (maximum or minimum points). When a function is strictly monotonic, its inverse will also be monotonic and will not have additional extremal points. This relationship is critical when determining the existence of extrema on the graph of an inverse function.

Inverse Function and Reflection

Reflecting a function’s graph about the line y = x produces the graph of its inverse function. Each point (a, b) on the original function corresponds to the point (b, a) on the inverse. This concept is fundamental when analyzing the properties of inverse functions, as it shows how inputs and outputs are interchanged while preserving the overall structure of the graph.

Tangent Lines and Their Orientation

Studying the tangent lines of a function involves examining the derivative to determine the slope at specific points on the curve. A horizontal tangent occurs where the derivative is zero, indicating a potential maximum or minimum, while a vertical tangent is present where the derivative is undefined or tends to infinity. Understanding the conditions for horizontal and vertical tangents is essential for analyzing the local behavior of functions and their inverses.

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