Question

Let f : → R, such that f’ (0) = 1 and f (x +2y) = f (x) + f (2y) + ex+2y (x + 2y) – x. ex – 2y. e2y + 4xy∀ xy Ïµ R. Find f (x).

Solution

Correct option is

We have, 

            

Replacing xy → 0, we get;   

           f (0) = f (0) + f (0) + 0 – 0 – 0 + 0 ⇒ f (0) = 0 

Replacing 2y → –x, we get;  

             

      

                     

  

                    

  

  

  

                     

Integrating (ii) both sides, 

       

                 

                

But    f (0) = 0 ⇒ c = 0 

So,    f (x) = x2 + x.ex

SIMILAR QUESTIONS

Q1

 for all real x and y. If f ’ (0) exists and equals to –1and f (0) = 1, find f ’(x).

Q2

Now if it is given that there exists a positive real δ, such that f (h) = h for 0 < h < δ then find f’(x) and hence f (x).

Q3

Let f  be an even function and f ’(0) exists, then find f’(0).

Q4

Let f (x) = xnn being non-negative integer. Then find the value of n for which the equality f’(a + b) = f ’(a) + f ’ (b) is valid for all, ab > 0

Q5

 

Q6

Find the set of points where x2 |x| is true thrice differentiable.

Q7

Find the number of points where f (x) = [sin x + cos x(where [.] denotes greatest integral function), x Ïµ [0, 2π] is not continuous.

Q8

  

differentiable function in [0, 2], find a and b. (where [.] denotes the greatest integer function).

Q9

Discuss the continuity of the function .

Q10

If g(x) is continuous function in [0, ∞) satisfying g(1) = 1. If

                          .