JEE Main · 2020 · 04 Sep 2020 · Shift 1 · Continuity and Differentiability

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 \to 0} \frac{f ( x )}{x} = 1

then

f^{'} (3)

is equal to ____ .

Answer: 10

Solution

Since

lim_{x \to 0} \frac{f ( x )}{x} = 1

exists,

f (0) = 0

. Writing the functional equation with increment

h

f (x + h) = f (x) + f (h) + x h^{2} + x^{2} h

, so

f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h} = lim_{h \to 0} (\frac{f ( h )}{h} + x h + x^{2}) = 1 + 0 + x^{2} = 1 + x^{2}

. Therefore

f^{'} (3) = 1 + 9 = 10

JEE Main · 2020 · 04 Sep 2020 · Shift 1 · Integer answer