The-possible-value-of-p-for-which-graph-of-the-function-f-x-2p-2-3ptan-x-tan-2-x-1-does-not-lie-below-x-axis-for-all-x-2-2-is-a-0-b-4-c-3-d-8-

Question Number 77024 by Sathvik last updated on 02/Jan/20

The possible value of p for which

graph of the function f (x) = {2 p}^{2} -

3 ptan x + \tan^{2} x + 1 does not lie below

x - axis for all x \in (\frac{- Π}{2}, \frac{Π}{2}) is

(a) 0 (b) 4 (c) 3 (d) 8

Answered by MJS last updated on 03/Jan/20

well, just test them all \dots

I found - 2 ⩽ p ⩽ 2

\Rightarrow (a) is the right answer

f (x) : y = \tan^{2} x - 3 p \tan x + 2 p^{2} + 1

f^{'} (x) : y = {2 \tan}^{3} x - 3 p \tan^{2} x + 2 \tan x - 3 p =

= (\tan^{2} x + 1) (2 \tan x - 3 p)

\Rightarrow \tan x = \frac{3}{2} p

insert in f (x)

y = 1 - \frac{p^{2}}{4} = 0 \Rightarrow p = \pm 2

\Rightarrow - 2 ⩽ p ⩽ 2

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