find-minimum-value-of-function-f-x-27-2x-2-96-27-x-2-

Question Number 111699 by bemath last updated on 04/Sep/20

f i n d m i n i m u m v a l u e o f f u n c t i o n

f (x) = \frac{27}{2 x^{2}} + \frac{96}{27} x^{2}

Answered by john santu last updated on 04/Sep/20

Answered by ajfour last updated on 04/Sep/20

f (x) = u + v

u v = c = 48

f (x) ∣_{m i n} = m n (\sqrt{{(u - v)}^{2} + 4 u v}) = 2 \sqrt{u v}

= 2 \sqrt{48} = 8 \sqrt{3} .

Commented by bemath last updated on 04/Sep/20

h o w y o u g o t u v = c = 48

Commented by ajfour last updated on 04/Sep/20

f (x) = u + v

u v = \frac{27}{2 x^{2}} \times \frac{96 x^{2}}{27} = 48

Answered by 1549442205PVT last updated on 04/Sep/20

Applying {Cauchy}^{'} s inequality for

positive numbers we have

f (x) = \frac{27}{2 x^{2}} + \frac{96}{27} x^{2} ⩾ 2 \sqrt{\frac{27}{{2 x}^{2}} . \frac{{96 x}^{2}}{27}}

= 2 \sqrt{48} = 8 \sqrt{3}

The equality ocurrs if and only if

\frac{27}{{2 x}^{2}} = \frac{{96 x}^{2}}{27} \Leftrightarrow x^{4} = \frac{27^{2}}{192} \Leftrightarrow x^{2} = \frac{27}{8 \sqrt{3}}

\Leftrightarrow∣ x ∣= \frac{3 \sqrt{3}}{\sqrt{8 \sqrt{3}}} \Leftrightarrow x = \pm \frac{3 \sqrt{3}}{\sqrt{8 \sqrt{3}}}

Thus, f (x) has smallest value equal to

8 \sqrt{3} when x = \pm \frac{3 \sqrt{3}}{\sqrt{8 \sqrt{3}}}