Question

Let f(x + y) = f(x) + f(y) + 2xy – 1 " x, y Î R. If f(x) is differentiable and f '(0) = sin f, then –

• A. f(x) > 0 " x Î R
• B. f(x) < 0 " x Î R
• C. f(x) = sin f" x Î R
• D. None of these

Answer 1

Answer: (A)

f(x) > 0 " x Î R

Answer 2

f(x + y) = f(x) + f(y) + 2xy – 1

Put x = y = 0 ̃ f(0) = 2f(0) – 1 ̃ f(0) = 1

f '(x) = $\operatorname { Lim } _ { h \rightarrow 0 }$ $\frac{f(x + h) - f(x)}{h}$

̃ $\operatorname { Lim } _ { h \rightarrow 0 }$ $\frac{f(x) + f(h) + 2xh - 1 - f(x)}{h}$

2x + $\operatorname { Lim } _ { h \rightarrow 0 }$ $\frac{f(h) - 1}{h}$ ̃ 2x + f '(0) = 2x + sinf

Integrating we gat f(x) = x² + x sin f + c

f (0) = 1 ̃ 1 = c \ f (x) = x² + x sin f + 1 > 0 " x Î R