Find where f(x)=4x−tan⁡x, −π2&lt;x&lt;π2 is increasing or decreasing and find it's maximum and minimum values.

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Find where f(x)=4x−tan⁡x, −π2&amp;lt;x&amp;lt;π2 is increasing or decreasing and find it&#039;s maximum and minimum values.

Nola Robson · Accepted Answer

f(x)=4x−tan⁡x, −π2&amp;lt;x&amp;lt;π2differentiating f(x) with respect to x, f′(x)=4−sec2xnow, ,f′(x)=4−sec2x=0⇒4−sec2x=0⇒sec2x=22⇒sec⁡x=±2⇒x=±π3so, there are three intervals.(−π2,π3),(−π3,π3),(π3,π2) let&#039;s check f&#039;(x)&amp;gt;0 in which intervals. −π3sec⁡x&amp;lt;2 so, f′(x)=4−sec2x&amp;gt;0hence,f(x)is increasing in(−π3,π3)similarly,check f′(x)value∈(−π2,−π3)and(π3,π2)you will get f′(x)&amp;lt;0 from this intervalsso,f(x)is decreasing∈(−π2,−π3)∪(π3,π2)again differentiate with respect to x, f″(x)=0−2sec2x.tan⁡x=−2sec2x.tan⁡xatx=π3,fπ3)=2sec2(π3).tan⁡(π3)&amp;lt;0hence,f(x)is maximum at x=π3

Find where f(x)=4x-\tan x, \ -\pi/2<x<\pi/2 is increasing or decreasing and find it's maximum and minimum values

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