cmi-entrance 2013 QB5

cmi-entrance · India · ugmath 20 marks Stationary points and optimisation Prove an inequality using calculus-based optimisation
Consider the function $f ( x ) = a x + \frac { 1 } { x + 1 }$, where $a$ is a positive constant. Let $L =$ the largest value of $f ( x )$ and $S =$ the smallest value of $f ( x )$ for $x \in [ 0,1 ]$. Show that $L - S > \frac { 1 } { 12 }$ for any $a > 0$. 
Consider the function $f ( x ) = a x + \frac { 1 } { x + 1 }$, where $a$ is a positive constant. Let $L =$ the largest value of $f ( x )$ and $S =$ the smallest value of $f ( x )$ for $x \in [ 0,1 ]$. Show that $L - S > \frac { 1 } { 12 }$ for any $a > 0$.
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