Let f - R - R such that f x -2 y - f x - f 2 y -4 xy - - x - y - R and f 0-0-If I 1-01 f dx - I 2-10 f x dx- and I 3-11 f x dx- thenA- I 2- I 2B- I3-0C- I 1 I 3

The correct options are A I2 = I2 C I1 lt; I3 f(x)=x^2 ⇒ I_1 = 1/3 = I_2 and I_3 = 2/3 ...

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Byju's Answer

Standard XII

Mathematics

Algebra of Derivatives

Let f : R → R...

Question

Let f: R $\to$ R such that f(x + 2y) = f(x) + f(2y) + 4xy, $\forall$ x, y $ϵ$ R and f(0) = 0.

If $I_{1} = \int_{0}^{1} f (d x), I_{2} = \int_{- 1}^{0} f (x) d x, a n d I_{3} = \int_{- 1}^{1} f (x) d x, t h e n$

I₂ = I₂

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I₃ = 0

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I₁ < I₃

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I₂ > I₃

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Solution

The correct options are
A
I₂ = I₂

C
I₁ < I₃

$f (x) = x^{2}$ $\Rightarrow$ $I_{1}$ $= \frac{1}{3}$ $= I_{2} a n d I_{3}$ $= \frac{2}{3}$

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Let f: R → R such that f(x + 2y) = f(x) + f(2y) + 4xy, ∀ x, y ϵ R and f(0) = 0. If I1=∫10f(dx),I2=∫0−1f(x)dx,and I3=∫1−1f(x)dx,then

The correct options are A I2 = I2 C I1 < I3 f(x)=x2 ⇒ I1 =13 =I2 and I3 =23

Let f: R $\to$ R such that f(x + 2y) = f(x) + f(2y) + 4xy, $\forall$ x, y $ϵ$ R and f(0) = 0.

If $I_{1} = \int_{0}^{1} f (d x), I_{2} = \int_{- 1}^{0} f (x) d x, a n d I_{3} = \int_{- 1}^{1} f (x) d x, t h e n$

The correct options are
A
I₂ = I₂

C
I₁ < I₃

$f (x) = x^{2}$ $\Rightarrow$ $I_{1}$ $= \frac{1}{3}$ $= I_{2} a n d I_{3}$ $= \frac{2}{3}$