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Zorluk: Çok zorFunctions and Custom Symbol Operations
For all non-zero real numbers xx, the function ff satisfies the functional equation
f(x)+2f(−1x)=3xf(x) + 2f\left(-\frac{1}{x}\right) = 3x

Which of the following statements must be true for all x≠0x \neq 0? Select all such statements.

  1. f(−1)=3f(-1) = 3Cevap
  2. B
    f(x)=f(−1x)f(x) = f\left(-\frac{1}{x}\right) for all x≠0x \neq 0
  3. f(x)≤−22f(x) \leq -2\sqrt{2} for all x>0x > 0Cevap
  4. D
    f(x)+f(−x)=6xf(x) + f(-x) = 6x for all x≠0x \neq 0
  5. E
    f(2)=3f(2) = 3

Cevap

The statements stating that f(−1)=3f(-1) = 3 and that f(x)≤−22f(x) \leq -2\sqrt{2} for all x>0x > 0 are the correct choices.
Solving the functional equation yields f(x)=−x−2xf(x) = -x - \frac{2}{x}. Substituting x=−1x = -1 gives f(−1)=3f(-1) = 3, making the statement asserting f(−1)=3f(-1) = 3 correct. Furthermore, for any positive real number xx, the sum x+2x≥22x + \frac{2}{x} \geq 2\sqrt{2} by the AM-GM inequality, so f(x)=−(x+2x)≤−22f(x) = -\left(x + \frac{2}{x}\right) \leq -2\sqrt{2}, making the statement asserting f(x)≤−22f(x) \leq -2\sqrt{2} for all x>0x > 0 correct.

Adım Adım Çözüm

1
Set up a system of functional equations by substituting x→−1xx \to -\frac{1}{x}.
Replacing xx with −1x-\frac{1}{x} in the original equation f(x)+2f(−1x)=3xf(x) + 2f\left(-\frac{1}{x}\right) = 3x gives f(−1x)+2f(x)=3(−1x)=−3xf\left(-\frac{1}{x}\right) + 2f(x) = 3\left(-\frac{1}{x}\right) = -\frac{3}{x}.
This creates a system of two linear algebraic equations with two unknowns: f(x)f(x) and f(−1x)f\left(-\frac{1}{x}\right).
2
Solve the system of equations for f(x)f(x).
Multiply the second equation by 22 to obtain 4f(x)+2f(−1x)=−6x4f(x) + 2f\left(-\frac{1}{x}\right) = -\frac{6}{x}. Subtracting the first equation f(x)+2f(−1x)=3xf(x) + 2f\left(-\frac{1}{x}\right) = 3x from this gives 3f(x)=−3x−6x3f(x) = -3x - \frac{6}{x}, so f(x)=−x−2xf(x) = -x - \frac{2}{x}.
Eliminating the composite term f(−1x)f\left(-\frac{1}{x}\right) yields an explicit expression for f(x)f(x).
3
Evaluate f(−1)f(-1).
f(−1)=−(−1)−2−1=1+2=3f(-1) = -(-1) - \frac{2}{-1} = 1 + 2 = 3.
This confirms that the statement asserting f(−1)=3f(-1) = 3 is correct.
4
Analyze the maximum value of f(x)f(x) for x>0x > 0 using the AM-GM inequality.
For x>0x > 0, x+2x≥2x⋅2x=22x + \frac{2}{x} \geq 2\sqrt{x \cdot \frac{2}{x}} = 2\sqrt{2}. Therefore, f(x)=−(x+2x)≤−22f(x) = -\left(x + \frac{2}{x}\right) \leq -2\sqrt{2}.
This confirms that the statement asserting f(x)≤−22f(x) \leq -2\sqrt{2} for all x>0x > 0 is correct.

Anahtar Kavram

Solving functional equations via variable substitution and analyzing function bounds via the AM-GM inequality.
Tahmini Süre:2m 30s
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Konular

Functions and Custom Symbol Operations

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