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(D) The center of mass $x_{cm}$ is given by the formula:$x_{cm} = \frac{\int x dm}{\int dm} = \frac{\int_{0}^{L} x \cdot k(x/L)^n dx}{\int_{0}^{L} k(x

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(D) The center of mass $x_{cm}$ is given by the formula:$x_{cm} = \frac{\int x dm}{\int dm} = \frac{\int_{0}^{L} x \cdot k(x/L)^n dx}{\int_{0}^{L} k(x/L)^n dx} = \frac{\int_{0}^{L} x^{n+1} dx}{\int_{0}^{L} x^n dx}$Evaluating the integrals:$x_{cm} = \frac{L^{n+2} / (n+2)}{L^{n+1} / (n+1)} = L \left( \frac{n+1}{n+2} ight)$Analyzing the behavior:$1$. When $n = 0$,$x_{cm} = L(1/2) = L/2$.$2$. As $n 	o \infty$,$x_{cm} = L \left( \frac{1 + 1/n}{1 + 2/n} ight) 	o L$.$3$. The derivative $\frac{dx_{cm}}{dn} = L \left( \frac{(n+2) - (n+1)}{(n+2)^2} ight) = \frac{L}{(n+2)^2} > 0$,which means $x_{cm}$ is an increasing function of $n$.Thus,the graph starts at $L/2$ for $n=0$ and asymptotically approaches $L$ as $n$ increases. This corresponds to the graph shown in option $D$.

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