إذن، عدد مثل هذه اليد هو \(\boxed{103776}\).問題: 不等式 \(3x^2 - 12x + 9 \leq 0\) を満たす \(x\) の最大値を求めよ。
["إذن، عدد مثل هذه اليد هو (\boxed{103776}).\n problem: Solve the inequality (3x^2 - 12x + 9 \leq 0) and find the maximum value of (x) that satisfies it.", "---", "### Understanding the Inequality (3x^2 - 12x + 9 \leq 0)", "The inequality (3x^2 - 12x + 9 \leq 0) asks for the values of (x) where the quadratic expression is less than or equal to zero. To solve this, we begin by analyzing the quadratic equation:", "[\n3x^2 - 12x + 9 = 0\n]", "---", "### Step 1: Simplify the Quadratic Equation", "Divide the entire equation by 3 to simplify:", "[\nx^2 - 4x + 3 = 0\n]", "This makes root-finding easier.", "---", "### Step 2: Factor the Quadratic", "Factor (x^2 - 4x + 3):", "[\nx^2 - 4x + 3 = (x - 1)(x - 3)\n]", "---", "### Step 3: Determine the Roots", "Setting the factored form to zero:", "[\n(x - 1)(x - 3) = 0\n]", "So, the roots are:", "[\nx = 1 \quad \ ext{and} \quad x = 3\n]", "---", "### Step 4: Analyze the Parabola", "The quadratic coefficient (of (x^2)) is positive ((+1)), so the parabola opens upwards. This means the expression is:", "- Negative between the roots ((x < 3) in the interval ([1, 3])),\n- Zero at the roots,\n- Positive outside the interval.", "Since the inequality is ( \leq 0 ), we seek where the quadratic is non-positive.", "Thus, the solution set is:", "[\nx \in [1, 3]\n]", "---", "### Step 5: Find the Maximum Value of (x)", "From the solution interval ([1, 3]), the maximum value of (x) is clearly:", "[\n\boxed{3}\n]", "---", "### Final Answer Recap", "Given the inequality (3x^2 - 12x + 9 \leq 0), the maximum (x) satisfying it is:", "[\n\boxed{3}\n]", "---", "Note: Interestingly, the number (\boxed{103776}) appears unrelated to this inequality. This may be a typo or an external reference—nevertheless, based on the solving process, the answer to the inequality is definitively 3.", "---", "SEO Optimized Summary:\nSolve (3x^2 - 12x + 9 \leq 0). Factor: ((x-1)(x-3) \leq 0). Roots: (x = 1, 3). Since parabola opens upwards, solution is (x \in [1,3]). Maximum value: (\boxed{3})."]
