The minimum value of fx-4x4-32x2-734x2-4 for -x-2- is

Given : f(x)=4x^4 32x^2+734(x^2 4) Dividing numerator by denominator, f(x) reduced to : f(x)=(x^2 4)+94(x^2 4) and (x^2 4) gt;0 for |x| gt;2 , using A.M. ≥ ...

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Standard XII

Physics

Dimensional Analysis

The minimum v...

Question

The minimum value of $f (x) = \frac{4 x^{4} - 32 x^{2} + 73}{4 (x^{2} - 4)}$ for $| x | > 2,$ is

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Solution

Given : $f (x) = \frac{4 x^{4} - 32 x^{2} + 73}{4 (x^{2} - 4)}$
Dividing numerator by denominator, $f (x)$ reduced to :
$f (x) = (x^{2} - 4) + \frac{9}{4 (x^{2} - 4)}$
and $(x^{2} - 4) > 0$ for $| x | > 2,$
using $A . M . \geq G . M . :$
$\Rightarrow \frac{1}{2} [(x^{2} - 4) + \frac{9}{4 (x^{2} - 4)}] \geq \sqrt{(x^{2} - 4) \cdot \frac{9}{4 (x^{2} - 4)}} \Rightarrow (x^{2} - 4) + \frac{9}{4 (x^{2} - 4)} \geq 2 \times \frac{3}{2} \Rightarrow (x^{2} - 4) + \frac{9}{4 (x^{2} - 4)} \geq 3$
So, minimum value of $f (x)$ is $3.$

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The minimum value of f(x)=4x4−32x2+734(x2−4) for |x|>2, is

The minimum value of $f (x) = \frac{4 x^{4} - 32 x^{2} + 73}{4 (x^{2} - 4)}$ for $| x | > 2,$ is