Let f( x )=∫ limits3 x ( dt /√ t 4+3 t 2+13). If g ( x ) is the inverse of f( x ) then g prime(0) has the value equal to

Question

Let f( x )=∫ limits3 x ( dt /√ t 4+3 t 2+13). If g ( x ) is the inverse of f( x ) then g prime(0) has the value equal to (A) (1/11) (B) 11 (C) √13 (D) (1/√13)

Tardigrade Admin · Accepted Answer

f prime( x ) ( dy / dx )=(1/√ x 4+3 x 2+13) ∴ g prime( y )=(1/ dy / dx )=√ x 4+3 x 2+13 when y=f(x) when y=0 then x=3 hence g prime(0)=√34+27+13=√121=11.

Q. Let $f (x) = 3 \int x \frac{d t}{t ^{4} + 3 t ^{2} + 13}$ . If $g (x)$ is the inverse of $f (x)$ then $g^{'} (0)$ has the value equal to

$\frac{1}{11}$

11

$13$

$\frac{1}{13}$

$f^{'} (x) \frac{d y}{d x} = \frac{1}{x ^{4} + 3 x ^{2} + 13}$
$∴ g^{'} (y) = \frac{1}{d y / d x} = x^{4} + 3 x^{2} + 13$
when $y = f (x)$
when $y = 0$ then $x = 3$
hence $g^{'} (0) = 3^{4} + 27 + 13 = 121 = 11$ .

Q. Let f(x)=3∫x​t4+3t2+13​dt​. If g(x) is the inverse of f(x) then g′(0) has the value equal to

111​

11

13​

13​1​

f′(x)dxdy​=x4+3x2+13​1​ ∴g′(y)=dy/dx1​=x4+3x2+13​ when y=f(x) when y=0 then x=3 hence g′(0)=34+27+13​=121​=11.

Q. Let $f (x) = 3 \int x \frac{d t}{t ^{4} + 3 t ^{2} + 13}$ . If $g (x)$ is the inverse of $f (x)$ then $g^{'} (0)$ has the value equal to

$\frac{1}{11}$

$13$

$\frac{1}{13}$

$f^{'} (x) \frac{d y}{d x} = \frac{1}{x ^{4} + 3 x ^{2} + 13}$
$∴ g^{'} (y) = \frac{1}{d y / d x} = x^{4} + 3 x^{2} + 13$
when $y = f (x)$
when $y = 0$ then $x = 3$
hence $g^{'} (0) = 3^{4} + 27 + 13 = 121 = 11$ .