Let f( x )= x + sin x. Suppose g denotes the inverse function of f. The value of g prime((π/4)+(1/√2)) has the value equal to

Question

Let f( x )= x + sin x. Suppose g denotes the inverse function of f. The value of g prime((π/4)+(1/√2)) has the value equal to (A) √2-1 (B) (√2+1/√2) (C) 2-√2 (D) √2+1

Tardigrade Admin · Accepted Answer

f(x)=y=x+ sin x (d y/d x)=1+ cos x g prime(y)=(d x/d y)=(1/1+ cos x) text where y=(π/4)+(1/√2)=x+ sin x ⇒ x=(π/4) ∴ g prime((π/4)+(1/√2))=(1/1+(1 / √2))=(√2/√2+1)=√2(√2-1)=2-√2

Q. Let $f (x) = x + sin x$ . Suppose $g$ denotes the inverse function of $f$ . The value of $g^{'} (\frac{π}{4} + \frac{1}{2})$ has the value equal to

$2 - 1$

$\frac{2 + 1}{2}$

$2 - 2$

$2 + 1$

$f (x) = y = x + sin x$
$\frac{d y}{d x} = 1 + cos x$
$g^{'} (y) = \frac{d x}{d y} = \frac{1}{1 + c o s x}$
$where y = \frac{π}{4} + \frac{1}{2} = x + sin x \Rightarrow x = \frac{π}{4}$
$∴ g^{'} (\frac{π}{4} + \frac{1}{2}) = \frac{1}{1 + ( 1/ 2 )} = \frac{2}{2 + 1} = 2 (2 - 1) = 2 - 2$

Q. Let f(x)=x+sinx. Suppose g denotes the inverse function of f. The value of g′(4π​+2​1​) has the value equal to

2​−1

2​2​+1​

2−2​

2​+1