Question
Consider function f(x) satisfies the relation, 
and differentiable for all x.
- If both the statements are TRUE and STATEMENT 2 is the correct explanation of STATMENT 1
- If both the statements are TRUE but STATEMENT 2 is not the correct explanation of STATMENT 1
- If STATEMENT 1 is true and STATEMENT 2 is false
- If STATEMENT 1 is false and STATEMENT 2 is true
Hint:
In this question, we have given a function f(x+y3) = f(x)+f(y3). Here two statements are given. It is like assertion and reason. Statement1 is assertion and statement 2 is reason, Find the statement 1 is correct or not and the statement 2 correct or not if correct then is its correct explanation.
The correct answer is: If both the statements are TRUE and STATEMENT 2 is the correct explanation of STATMENT 1
Here we have to find the which given statement is correct.
Firstly, given is
Statement1: if f’(2) = a then f’(-2) =a
Statement 2: f(x) is an odd function
We have,
F(x) = f(x+y3) = f(x)+f(y3) ∀ x, y ϵ R
Put x=y=0, we get
f(0+0)=f(0)+f(0)⇒f(0) = 0.
Now, put y=
, we get
f(0) = f(x)+f(−x)
⇒f(x)+f(−x) = 0
⇒f(x) is an odd function
⇒f′(x) is an even function
⇒f’(−2) = a
Therefore, the statement 1 and statement 2 is correct and statement 2 is correct explanation.
The correct answer is If both the statement is TRUE and STATEMENT 2 is correct explanation of STATEMENT 1.
In this question, we have given two statements are like assertion and reason. The odd function is, a function such that f (−x) =−f (x) where the sign is reversed but the absolute value remains the same if the sign of the independent variable is reversed.
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