If f(x) be differentiable function and curve y=f(x) passes through (1,1) and satisfies the relation 2 f(x+y)+f(x-y)+3 y2=3 f(x)+2 x y, then displaystyle lim x arrow 1 (f(x)-1/x-1) is equal to

Question

If f(x) be differentiable function and curve y=f(x) passes through (1,1) and satisfies the relation 2 f(x+y)+f(x-y)+3 y2=3 f(x)+2 x y, then displaystyle lim x arrow 1 (f(x)-1/x-1) is equal to (A) 3 (B) 0 (C) 2 (D) 1

Tardigrade Admin · Accepted Answer

Put x=1,y=h in the relation 2((f (. 1 + h .) - f (. 1 .)/h))-((. f (. 1 - h .) - f (. 1 .) .)/- h)+3h=2 take limit h arrow 0 2f'(.1.)-f'(.1.)=2⇒ f'(.1.)=2

Q. If $f (x)$ be differentiable function and curve $y = f (x)$ passes through $(1, 1)$ and satisfies the relation $2 f (x + y) + f (x - y) + 3 y^{2} = 3 f (x) + 2 x y$ , then $x \to 1 lim \frac{f ( x ) - 1}{x - 1}$ is equal to

$3$

$0$

$2$

$1$

Put $x = 1, y = h$ in the relation $2 (\frac{f ( 1 + h ) - f ( 1 )}{h}) - \frac{( f ( 1 - h ) - f ( 1 ) )}{- h} + 3 h = 2$
take limit $h \to 0$
$2 f^{^{'}} (1) - f^{^{'}} (1) = 2 \Rightarrow f^{^{'}} (1) = 2$

Q. If f(x) be differentiable function and curve y=f(x) passes through (1,1) and satisfies the relation 2f(x+y)+f(x−y)+3y2=3f(x)+2xy, then x→1lim​x−1f(x)−1​ is equal to

3

0

2

1

Put x=1,y=h in the relation 2(hf(1+h)−f(1)​)−−h(f(1−h)−f(1))​+3h=2 take limit h→0 2f′(1)−f′(1)=2⇒f′(1)=2

Q. If $f (x)$ be differentiable function and curve $y = f (x)$ passes through $(1, 1)$ and satisfies the relation $2 f (x + y) + f (x - y) + 3 y^{2} = 3 f (x) + 2 x y$ , then $x \to 1 lim \frac{f ( x ) - 1}{x - 1}$ is equal to

$3$

$0$

$2$

$1$

Put $x = 1, y = h$ in the relation $2 (\frac{f ( 1 + h ) - f ( 1 )}{h}) - \frac{( f ( 1 - h ) - f ( 1 ) )}{- h} + 3 h = 2$
take limit $h \to 0$
$2 f^{^{'}} (1) - f^{^{'}} (1) = 2 \Rightarrow f^{^{'}} (1) = 2$