Suppose a differentiable function f x satisfies the identity f x + y = f x + f y + x y 2 + x 2 y , for all real x and y . If lim x → 0 f x x = 1 , then f ' 3 is equal to :

Question

Suppose a differentiable function f x satisfies the identity f x + y = f x + f y + x y 2 + x 2 y , for all real x and y . If lim x &#8594; 0 f x x = 1 , then f ' 3 is equal to :

Quantrex Academy · Accepted Answer

Correct answer: 0

&#8757; lim x &#8594; 0 f x x = 1 &#160; &#8658; f ' 0 = 1 f x + y = f x + f y + x y 2 + x 2 y Differentiate w.r.t. x keeping y constant f ' x + y = f ' x + 0 + y 2 + 2 x y put y = - x f ' 0 = f ' x + x 2 &#8722; 2 x 2 1 = f ' x &#8722; x 2 f ' x = 1 + x 2 f ' 3 = 10 .

Suppose a differentiable function f x satisfies the identity f x + y = f x + f y + x y 2 + x 2 y , for all real x and y . If lim x → 0 f x x = 1 , then f ' 3 is equal to :

Correct answer

Step-by-step solution

More from Functions

Suppose a differentiable function f x satisfies the identity f x + y = f x + f y + x y 2 + x 2 y , for all real x and y . If lim x &#8594; 0 f x x = 1 , then f &#39; 3 is equal to :

Correct answer

Step-by-step solution

More from Functions

Suppose a differentiable function f x satisfies the identity f x + y = f x + f y + x y 2 + x 2 y , for all real x and y . If lim x → 0 f x x = 1 , then f ' 3 is equal to :