STD 11 - 4.2 Quadratic equations and inequations — JEE कक्षा 11 साइन्स — Question

Let $f(x)=e^{2 x}\left(e^{2 x}+\frac{1}{e^{2 x}}+4\left(e^{x}+\frac{1}{e^{x}}\right)-58\right)$

$e^{x}+\frac{1}{e^{x}}$

Let $h(t)=t^{2}+4 t-58=0$

$t =\frac{-4 \pm \sqrt{16+4.58}}{2}$

$\frac{-4 \pm 2 \sqrt{62}}{2}$

$t _{1}=-2+2 \sqrt{62}$

$t _{2}=-2-2 \sqrt{62}$ (not possible)

$t \geq 2$

$e ^{ x }+\frac{1}{ e ^{ x }}=-2+2 \sqrt{62}$

$e ^{2 x }-(-2+2 \sqrt{62}) e ^{ x }+1=0$

$(-2+2 \sqrt{62})-4$

$4+4.62-8 \sqrt{62}-4$

$248-8 \sqrt{62}>0$

$\frac{- b }{2 a }>0$

both roots are positive

$2$ real roots