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

# Let f : R → R, Such That f’ (0) = 1 and f (x +2y) = f (x) + f (2y) + Ex+2y (x + 2y) – x. Ex – 2y. E2y + 4xy, ∀ x, y Ïµ R. Find f (x).

#### Video lectures

Access over 500+ hours of video lectures 24*7, covering complete syllabus for JEE preparation.

#### Online Support

Practice over 30000+ questions starting from basic level to JEE advance level.

#### National Mock Tests

Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation.

#### Organized Learning

Proper planning to complete syllabus is the key to get a decent rank in JEE.

#### Test Series/Daily assignments

Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation.

## Question

### 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

.