Question

Let f : $R\to R$ such that $f(x+2y)=f\left(x\right)+f\left(2y\right)+4xy,\forall x,yϵR$ and f(0) = 0.If $l_1={\int }_0^{1}f\left(x\right)dx,I_2={\int }_{-1}^{0}f\left(x\right)dx$, and $l_3={\int }_{-1}^{1}f\left(x\right)dx,$ then

• A. $I_1=I_2$
• B. $I_3=0$
• C. $I_1<I_3$
• D. $I_2<I_3$

Answer 1

Answer: (A), (C)

The correct options are
A

$I_1=I_2$

C

$I_1<I_3$

$f\left(x\right)=x^2\Rightarrow I_1=\frac{1}{3}{I}_{2}and{I}_3=\frac{2}{3}$