csat-suneung 2013 Q21

csat-suneung · South-Korea · csat__math-science 4 marks Stationary points and optimisation Composite or piecewise function extremum analysis

For the function $f ( x ) = k x ^ { 2 } e ^ { - x } ( k > 0 )$ and a real number $t$, let $g ( t )$ be the smaller of the distance from the point $( t , f ( t ) )$ on the curve $y = f ( x )$ to the $x$-axis and the distance to the $y$-axis. What is the maximum value of $k$ such that the function $g ( t )$ is not differentiable at exactly one point? [4 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { \sqrt { e } }$
(3) $\frac { e } { 2 }$
(4) $\sqrt { e }$
(5) $e$

For the function $f ( x ) = k x ^ { 2 } e ^ { - x } ( k > 0 )$ and a real number $t$, let $g ( t )$ be the smaller of the distance from the point $( t , f ( t ) )$ on the curve $y = f ( x )$ to the $x$-axis and the distance to the $y$-axis. What is the maximum value of $k$ such that the function $g ( t )$ is not differentiable at exactly one point? [4 points]\\
(1) $\frac { 1 } { e }$\\
(2) $\frac { 1 } { \sqrt { e } }$\\
(3) $\frac { e } { 2 }$\\
(4) $\sqrt { e }$\\
(5) $e$

Paper Questions

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