﻿ Let f be a real function satisfying f (x + y + z) = f (x) f (y) f (z) for all real x, y, z . If f (2) = 4 and f’ (0) = 3. Then find f (0) and f’ (2). : Kaysons Education

# Let f be A Real Function Satisfying f (x + y + z) = f (x) f (y) f (z) for All Real x, y, z . If f (2) = 4 And f’ (0) = 3. Then Find f (0) And f’ (2).

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## Question

### Solution

Correct option is

f (0) = 1 and f ’ (4) = 12

Here,

(x + y + z) = f (xf (yf (z) for all xyz Ïµ R        … (i)

put    x = y = z =

f (0) = (f (0))3 ⇒ f (0) = 0, ± 1                              …(ii)

putting        = – 1 in (i), we get

f (x – 2) = f (x) {f (– 1)}2

⇒      f (0) = f (2) {f (– 1)}2 for all Ïµ

⇒      f (0) = 4 {f (– 1)}2

⇒      (0) > 0                                                               … (iii)

∴   from (ii) and (iii),

f (0) = 1                                                              … (iv)

Now putting = 2 and z = 0 in (i). we get

f (x + 2) = (xf (2) (0)

f (x + 2) = 4 f (x

f’ (x + 2) = 4 f’ (x), putting x = 2

f’ (4) = 4, f’ (2) = 12

Thus,       f (0) = 1    and     f’ (4) = 12

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