For any non-zero real number x, let f(x)+2f(1/x)=3x
Then, the sum of all possible values of x for which f(x)=, is

Question

For any non-zero real number x, let f(x)+2f(1/x)=3x
 Then, the sum of all possible values of x for which f(x)=, is

UPSC Quizroom · Accepted Answer

Step 1: Substitute x → 1/x

Original equation:

f(x) + 2f(1/x) = 3x … (1)

Replace x by 1/x (x ≠ 0):

f(1/x) + 2f(x) = 3*(1/x) … (2)

Step 2: Solve the system for f(x) and f(1/x)

From (1): f(x) + 2f(1/x) = 3x

From (2): 2f(x) + f(1/x) = 3 / x

Let’s solve:

Multiply (2) by 2: 4f(x) + 2f(1/x) = 6 / x

Subtract (1): (4f(x) + 2f(1/x)) − (f(x) + 2f(1/x)) = 6/x − 3x

3f(x) = 6/x − 3x → f(x) = 2/x − x

Step 3: Solve f(x) = 3

f(x) = 2/x − x = 3

2
/
𝑥
−
𝑥
=
3
  
⟹
  
2
−
𝑥
2
=
3
𝑥
  
⟹
  
𝑥
2
+
3
𝑥
−
2
=
0
2/x−x=3⟹2−x
2
=3x⟹x
2
+3x−2=0

Solve quadratic: x² + 3x − 2 = 0

𝑥
=
[
−
3
±
√
(
9
+
8
)
]
/
2
=
[
−
3
±
√
17
]
/
2
x=[−3±√(9+8)]/2=[−3±√17]/2
Step 4: Sum of roots

Sum of roots = −(coefficient of x) / (coefficient of x²) = −3 / 1 = −3

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