Let f:R-R be a differential function satisfying f

Question

Let f:R-R be a differential function satisfyingf(x+y) + f(x).f(y) = f(xy+1).Also, f(0) = -1, f'(0)= f'(1)= 1. Then, a) f(1) = 0b) f(2) = f'(2)c) f(1) ≠ 0d) f(2) ≠ f'(2) Ans is (AB). Please explain how

Aditya Gupta · Accepted Answer

put x=0 and y=1 in given relation we getf(1) – f(1)= f(1) so that f(1)= 0.Now, f(x+y) – f(x) = f(xy+1) – f(x)*[f(y)+1]or Lt y → 0 [f(x+y) – f(x)]/y = Lt y → 0 f(xy+1)/y – f(x)*Lt y → 0[f(y)+1]/yor f’(x)= x*Lt y → 0 f(xy+1)/xy – f(x)*Lt y → 0[f(y) – f(0)]/(y – 0)for Lt y → 0 f(xy+1)/xy, put t= xy, so it becomes Lt t → 0 f(t+1)/t = Lt t → 0 [f(1+t) – f(1)]/t = f’(1)= 1so, we get f’(x)= x*1 – f(x)*f’(0)= x – f(x)let f(x)= zso dz/dx + z= xIF= e^xso d(z.e^x)/dx = xe^xor z.e^x= e^x.(x – 1) + Cwhen x=1, z=0, so C=0So z.e^x= e^x.(x – 1)or z= f(x)= x – 1.So, f(2)= 1, f’(x)= 1 so f’(2)= 1.or f(2) = f'(2) = 1.So options A, B.KINDLY APPROVE :D

Differential Calculus> Let f:R-R be a differential function sati...

Let f:R-R be a differential function satisfying
f(x+y) + f(x).f(y) = f(xy+1).
Also, f(0) = -1, f'(0)= f'(1)= 1. Then,

a) f(1) = 0
b) f(2) = f'(2)
c) f(1) ≠ 0
d) f(2) ≠ f'(2)

Ans is (AB). Please explain how

QuackLaLa , 5 Years ago

Aditya Gupta

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Differential Calculus> Let f:R-R be a differential function sati...

Let f:R-R be a differential function satisfyingf(x+y) + f(x).f(y) = f(xy+1).Also, f(0) = -1, f'(0)= f'(1)= 1. Then, a) f(1) = 0b) f(2) = f'(2)c) f(1) ≠ 0d) f(2) ≠ f'(2) Ans is (AB). Please explain how

QuackLaLa , 5 Years ago

Aditya Gupta

LIVE ONLINE CLASSES

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Let f:R-R be a differential function satisfying
f(x+y) + f(x).f(y) = f(xy+1).
Also, f(0) = -1, f'(0)= f'(1)= 1. Then,

a) f(1) = 0
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Ans is (AB). Please explain how

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Please give me the solution of this problem.

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where A + B + C = T
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