Question

Let f be a real function satisfying f (x + z) = f (xf (yf (zfor all real xyz . If f (2) = 4 and f’ (0) = 3. Then find f (0) and f’ (2).

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

SIMILAR QUESTIONS

Q1

Let f (x) = [n + p sin x], x Ïµ (0, π), n Ïµ Z and p is a prime number, where [.] denotes the greatest integer function. Then find the number of points where f (x) is not Differential.

Q2

, then draw the graph of f (x) in the interval [–2, 2] and discuss the continuity and differentiability in [–2, 2]

Q3

Fill in the blank, statement given below let . The set of points where f (x) is twice differentiable is ……………. .

Q4

The function f (x) = (x2 – 1) |x2 – 3x +2| + cos ( | | ) is not differentiable at

Q5
Q6

The number of points in (1, 3), where is not differentiable is:

Q7

Let f and g be differentiable function satisfying g’ (a) = 2, g (a) = b and fog = I (identity function) Then, f’(b) is equal to:

Q8

If the function , (where [.] denotes the greatest integer function) is continuous and differentiable in (4, 6), then.

Q9

Let [.] denotes the greatest integer function and f (x) = [tan2x], then:

Q10

Let h(x) = min.{xx2} for every real number of x. Then: