Find-max-amp-min-of-function-f-x-4tan-x-3cot-x-

Question Number 142167 by iloveisrael last updated on 27/May/21

F i n d m a x & m i n o f f u n c t i o n

f (x) = 4 \tan x + 3 \cot x

Answered by MJS_new last updated on 27/May/21

- \infty < \tan x < + \infty \Rightarrow no absolute \min, \max

g (t) = 4 t + \frac{3}{t}

g^{'} (t) = 4 - \frac{3}{t^{2}} = 0 \Leftrightarrow 4 t^{2} - 3 = 0 \Leftrightarrow t = \pm \frac{\sqrt{3}}{2}

g^{″} (t) = \frac{6}{t^{3}}

g^{″} (- \frac{\sqrt{3}}{2}) < 0 \Rightarrow g (t) has a local \max at (- \frac{\sqrt{3}}{2}, - 4 \sqrt{3})

g^{″} (\frac{\sqrt{3}}{2}) > 0 \Rightarrow g (t) has a local \min at (\frac{\sqrt{3}}{2}, 4 \sqrt{3})

\Rightarrow

f (x) has local maxima when \tan x = - \frac{\sqrt{3}}{2}

f (x) has local minima when \tan x = \frac{\sqrt{3}}{2}

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