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Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ y_0>0,\,y_0 <f(x_0).$ Light is assumed to be reflected from the plot $y=f(x)$ and the $x$-axis. Does there exist a number $R$ s.t. the part of the graph $y=f(x)$ for $x>R$ is not lightened?

The model example $f(x):=e^{-x},\,P(0,0.5)$ suggests the answer is yes.

The question is migrated from SE.

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ y_0>0,\,y_0 <f(x_0).$ Light is assumed to be reflected from the plot $y=f(x)$ and the $x$-axis. Does there exist a number $R$ s.t. the part of the graph $y=f(x)$ for $x>R$ is not lightened?

The model example $f(x):=e^{-x},\,P(0,0.5)$ suggests the answer is yes.

The question is migrated from SE.

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}_+$ (the positive half-ray) and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ y_0>0,\,y_0 <f(x_0).$ Light is assumed to be reflected from the plot $y=f(x)$ and the $x$-axis. Does there exist a number $R$ s.t. the part of the graph $y=f(x)$ for $x>R$ is not lightened?

The model example $f(x):=e^{-x},\,P(0,0.5)$ suggests the answer is yes.

The question is migrated from SE.

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ y_0>0,\,y_0 <f(x_0).$ Light is assumed to be reflected from the plot $y=f(x)$ and the $x$-axis. Does there exist a number $R$ s.t. the part of the graph $y=f(x)$ for $x>R$ is not lightened?

The model example $f(x):=e^{-x},\,P(0,0.5)$ suggests the answer is yes.

The question is migrated from SE.