If fx-y-fx-fy-2 x y-6 for all x- y - R and f0-2- then y-fx will be

The correct option is B parabolaf'(x) = n → 0limf(x+h) f(x+0)/h = n → 0limf(x) + f(h) + 2xh 6 f(x) f(0) + 6/h = 2x + n → 0limf(0+h) f(0)/h = 2x + f'( ...

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Byju's Answer

Standard XII

Mathematics

Definition of Circle

If fx+y=fx+fy...

Question

If f(x + y) = f(x) + f(y) + 2xy - 6 for all $x, y ϵ R$ and f'(0) = 2, then y = f(x) will be

straight line

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parabola

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ellipse

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circle

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Solution

The correct option is B parabola
$f^{'} (x) = l i m n \to 0 \frac{f (x + h) - f (x + 0)}{h} = l i m n \to 0 \frac{f (x) + f (h) + 2 x h - 6 - f (x) - f (0) + 6}{h} = 2 x + l i m n \to 0 \frac{f (0 + h) - f (0)}{h} = 2 x + f^{'} (0) f (x) = x^{2} + 2 x + 6$
So, $f (x) = x^{2} + 2 x + 6.$

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Similar questions

Q. If f(x+y)=f(x) + f(y) + 2xy - 6 for all

x, y ϵ R

and f'(0) = 2, then y = f(x) will be

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f (x + y) = f (x) + f (y) + 2 x y - 1 \forall x, y \in R .

f (x)

is differentiable for

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f^{'} (0) = sin ϕ

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for all real

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f^{'} (0) = cos α

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\forall x \in R

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If f(x + y) = f(x) + f(y) + 2xy - 6 for all x,yϵR and f'(0) = 2, then y = f(x) will be

The correct option is B parabolaf′(x)=limn→0f(x+h)−f(x+0)h=limn→0f(x)+f(h)+2xh−6−f(x)−f(0)+6h=2x+limn→0f(0+h)−f(0)h=2x+f′(0)f(x)=x2+2x+6 So, f(x)=x2+2x+6.

If f(x + y) = f(x) + f(y) + 2xy - 6 for all $x, y ϵ R$ and f'(0) = 2, then y = f(x) will be

The correct option is B parabola
$f^{'} (x) = l i m n \to 0 \frac{f (x + h) - f (x + 0)}{h} = l i m n \to 0 \frac{f (x) + f (h) + 2 x h - 6 - f (x) - f (0) + 6}{h} = 2 x + l i m n \to 0 \frac{f (0 + h) - f (0)}{h} = 2 x + f^{'} (0) f (x) = x^{2} + 2 x + 6$
So, $f (x) = x^{2} + 2 x + 6.$