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Derivative of Standard Functions

let f(x) be a...

Question

let f(x) be a function such that f(x).f(y)=f(x+y),f(0)=1,f(1)=4.If 2g(x)=f(x)(1-g(x)) then find g(x)+g(1-x)

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Solution

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f (x)

f (x) . f (y) = f (x + y), f (0) = 1, f (1) = 4

2 g (x) = f (x) (1 - g (x))

f (x)

g (x)

0 \leq x \leq 1

f (0) = 10

g (0) = 2

f (1) = 2

g (1) = 4

(0, 1)

f (x), g (x)

f (1) = g (1) = 2

lim x \to 1 \frac{f (1) g (x) - f (x) g (1) - f (1) + g (1)}{g (x) - f (x)}

f (x), g (x)

f (1) = g (1) = 2

lim x \to 1 \frac{f (1) g (x) - f (x) g (1) - f (1) + g (1)}{g (x) - f (x)}

f (x)

g (x)

0 \leq x \leq 1

f (0) = 2, g (0) = 0, f (1) = 6

[0, 1]

f^{'} (c) = 2 g^{'} (c)

g (1) =

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