Let f: R arrow R be a function defined as f ( x )= e x 2( e x 3+ x +1- e - x 3- x -3)+2 x +5 and g is the inverse function of f, then

Question

Let f: R arrow R be a function defined as f ( x )= e x 2( e x 3+ x +1- e - x 3- x -3)+2 x +5 and g is the inverse function of f, then (A) (d/d x)(x f(g(x)))|x=5=10 (B) (d/d x)(g(x))|x=3=(1/10) (C) .( d / dx )(( x / g ( x )))| x =3=-31 (D) .( d / dx )(( g ( f ( x )))/ x 2))| x = e =1

Tardigrade Admin · Accepted Answer

f ( x )= e x 3+ x 2+ x +1- e - x 3+ x 2- x -3+2 x +5 f (-1)=3 f prime( x )= e x 3+ x 2+ x +1(3 x 2+2 x +1)- e - x 3+ x 2- x -3(-3 x 2+2 x -1)+2 f prime(-1)=3-2+1-(-3-2-1)+2=10 text Now, verify the options.

Q. Let $f : R \to R$ be a function defined as $f (x) = e^{x^{2}} (e^{x^{3} + x + 1} - e^{- x^{3} - x - 3}) + 2 x + 5$ and $g$ is the inverse function of $f$ , then

$\frac{d}{d x} (x f (g (x))) ∣_{x = 5} = 10$

$\frac{d}{d x} (g (x)) ∣_{x = 3} = \frac{1}{10}$

$\frac{d}{d x} (\frac{x}{g ( x )}) ∣ ∣_{x = 3} = - 31$

$\frac{d}{d x} (\frac{g ( f ( x )))}{x ^{2}}) ∣ ∣_{x = e} = 1$

$f (x) = e^{x^{3} + x^{2} + x + 1} - e^{- x^{3} + x^{2} - x - 3} + 2 x + 5$
$f (- 1) = 3$
$f^{'} (x) = e^{x^{3} + x^{2} + x + 1} (3 x^{2} + 2 x + 1) - e^{- x^{3} + x^{2} - x - 3} (- 3 x^{2} + 2 x - 1) + 2$
$f^{'} (- 1) = 3 - 2 + 1 - (- 3 - 2 - 1) + 2 = 10$
$Now, verify the options.$

Q. Let f:R→R be a function defined as f(x)=ex2(ex3+x+1−e−x3−x−3)+2x+5 and g is the inverse function of f, then

dxd​(xf(g(x)))∣x=5​=10

dxd​(g(x))∣x=3​=101​

dxd​(g(x)x​)∣∣​x=3​=−31

dxd​(x2g(f(x)))​)∣∣​x=e​=1