Use the derivative test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ \frac { k ^ { 2 } } { e ^ { k } } \right\}$
A) Strictly decreasing
B) Strictly increasing
C) Eventually decreasing
D) Eventually increasing

Question

Quizplus · Accepted Answer

To analyze the monotonicity of the sequence $\left\{ \frac { k ^ { 2 } } { e ^ { k } } \right\}$, we consider its continuous counterpart $f(x) = \frac{x^2}{e^x}$ and differentiate it. The first derivative is $f'(x) = \frac{2x}{e^x} - \frac{x^2e^x}{e^{2x}} = \frac{2x - x^2}{e^x}$. Setting $f'(x) = 0$ gives $x = 2$ as a critical point. Further analysis or the second derivative test shows that the function is increasing for $x < 2$ and decreasing for $x > 2$, implying the sequence is eventually decreasing.

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