isi-entrance 2015 QB6

isi-entrance · India · UGA Stationary points and optimisation Find critical points and classify extrema of a given function
Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.
The function $f ( x ) - 4$ is a sum of squares and hence non-negative. So the minimum is 4 which is attained at $x = 1$. 
Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained?
$$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$
Sketch on plain paper the graph of this function.
Paper Questions
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