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f(x)f(y)=f(x)+f(y)+f(xy)−2f\left( x \right) f\left( y \right) =f\left( x \right) +f\left( y \right) +f\left( xy \right) -2f(x)f(y)=f(x)+f(y)+f(xy)−2
If f(x)f\left( x \right) f(x) is a polynomial satisfying the equation above for all xxx and yyy and f(2)=5f\left( 2 \right)=5 f(2)=5, then compute limx→2f′(x)\displaystyle \lim _{ x\rightarrow 2 }{ f^{ ' }\left( x \right) } x→2limf′(x).
If anyone solves this one, please do tell me what is the error in this one
Note by Akhilesh Prasad 5 years, 2 months ago
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Hey there! We have, f(x)f(y)−f(x)−f(y)+1=f(xy)−1(f(x)−1)(f(y)−1)=f(xy)−1g(x)=f(x)−1⇒g(x)g(y)=g(xy) f(x)f(y)-f(x)-f(y)+1\quad =\quad f(xy)-1\\ (f(x)-1)(f(y)-1)\quad =\quad f(xy)-1\quad \\ g(x)=f(x)-1 \\ \Rightarrow g(x)g(y) = g(xy) \\ f(x)f(y)−f(x)−f(y)+1=f(xy)−1(f(x)−1)(f(y)−1)=f(xy)−1g(x)=f(x)−1⇒g(x)g(y)=g(xy)
This is famous, see here. We then have that, g(x)=xt,t=2.g(x)=x2f(x)=x2+1f′(x)=2x⇒limx→2f′(x)=4 g(x) = x ^ t, t = 2. \\ g(x) = x^2 \\ f(x) = x^2 + 1 \\ f'(x) = 2x \\ \Rightarrow \lim _{ x\rightarrow 2 }{ f^{ ' }(x) } = 4 g(x)=xt,t=2.g(x)=x2f(x)=x2+1f′(x)=2x⇒limx→2f′(x)=4
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@Rishabh Cool, this question again.
@parv mor
Put x and y as 2 which will give you the value of f(4) as 17
Put x as 4 and y as 2 and you get f(8) as 65. Similarly you can check with other values and this leads to the conclusion that f(x) is one greater than the square of x.
Hence the limit computation will be 4
Please excuse me if you were looking for a formal proof or evaluation of the same.
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Top NewestHey there!
We have,
f(x)f(y)−f(x)−f(y)+1=f(xy)−1(f(x)−1)(f(y)−1)=f(xy)−1g(x)=f(x)−1⇒g(x)g(y)=g(xy)
This is famous, see here.
We then have that,
g(x)=xt,t=2.g(x)=x2f(x)=x2+1f′(x)=2x⇒limx→2f′(x)=4
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@Rishabh Cool, this question again.
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@parv mor
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Put x and y as 2 which will give you the value of f(4) as 17
Put x as 4 and y as 2 and you get f(8) as 65. Similarly you can check with other values and this leads to the conclusion that f(x) is one greater than the square of x.
Hence the limit computation will be 4
Please excuse me if you were looking for a formal proof or evaluation of the same.
Log in to reply