How do you find the maximum, minimum and inflection points and concavity for the function $f (x) = x^{2} + \frac{1}{x^{2}}$ $f(x) = x^2 + 1/x^2$ ?

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Answer 1 · 2017-01-12T03:40:41

$f \geq 2$ $f >=2$ . Symmetrical about y-axis.
Minimum : 2. No maximum. No POI ( point of inflexion ).
Convex towards origin.

$f \to \pm \infty$ $f to +-oo$ , as $x \to 0 and \pm \infty$ $x to 0 and +-oo$ .

$f > 0$ $f > 0$ , with the minimum given by

$f'=2x-2/x^3=0$ , giving zeros x=+-1#.

The minimum f is $f(+-1)=2$ , twice.

$f''=2+6/x^4 > 0$ .

$x = 0 uarr$ is the asymptote.

On the left ( $Q_2$ ) and the right of the asymptote, ( $Q_1$ ),

f' is increasing, from $-oo to oo$ , revealing convexity,

in respect of each branch of the graph, in $Q_1 and Q_2$ ..

So, f is minimum 2 at $x = +-1$ and there is no point of inflexion.

graph{(y-2)(y-x^2-1/x^2)=0 [-20, 20, -10, 10]}

How do you find the maximum, minimum and inflection points and concavity for the function f(x) = x^2 + 1/x^2f(x)=x2+1x2f(x) = x^2 + 1/x^2?