FunctionMedium
Question
Given the function f(x) such that 2f(x) + xf 
- 2f 
= 4 cos2 
+ x cos 
, then which one of the following is correct ?
- 2f = 4 cos2 + x cos , then which one of the following is correct ?Options
A.f(2) + f(1/2) = 1
B.f(1) = -1 but the values of f(2), f(1/2) cannot be determined
C.f(2) + f(1) = f(1/2)
D.f(2) + f(1) = 0
Solution
put x = 1
2f(1) + 1 f(1) - 2f( | √2 sin
| ) = -1
⇒ 3 f (1) - 2 f (1) = -1 ⇒ f (1) = -1
Now put x = 2
2 f(2) + 2 f
- 2f (1) = 4 cos2 π + 2 cos 
⇒ 2 f(2) + 2f
- 2 f (1) = 4
⇒ f(2) + f
= 1 ......... (i)
Now put x = 1/2 we get
4f
+ f (2) = 1 .............. (ii)
from (i) and (ii)
f
= 0 & f (2) = 1
2f(1) + 1 f(1) - 2f( | √2 sin
| ) = -1⇒ 3 f (1) - 2 f (1) = -1 ⇒ f (1) = -1
Now put x = 2
2 f(2) + 2 f
- 2f (1) = 4 cos2 π + 2 cos ⇒ 2 f(2) + 2f
- 2 f (1) = 4⇒ f(2) + f
= 1 ......... (i)Now put x = 1/2 we get
4f
+ f (2) = 1 .............. (ii)from (i) and (ii)
f
= 0 & f (2) = 1Create a free account to view solution
View Solution FreeMore Function Questions
If y = f(x), then...If f (x) = |x| + |x − 1|, then for 0 < x < 1, f (x) equals-...Let f(x) = x (2 - x), 0 ≤ x ≤ 2. If the definition of ′f′ is extended over the set ,R - [0 , 2] ...Given below are two statements:Statement I: The function $f:\mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x) = \frac{...Let f : R → R be a function defined by f(x) = x3 + x2 + 3x + sin x. Then f is:...