{
  "index": "1986-A-1",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the\nset of all real numbers $x$ satisfying $x^4+36\\leq 13x^2$.",
  "solution": "Solution. The condition \\( x^{4}+36 \\leq 13 x^{2} \\) is equivalent to \\( (x-3)(x-2)(x+2)(x+3) \\leq \\) 0 , which is satisfied if and only if \\( x \\in[-3,-2] \\cup[2,3] \\). The function \\( f \\) is increasing on \\( [-3,-2] \\) and on \\( [2,3] \\), since \\( f^{\\prime}(x)=3\\left(x^{2}-1\\right)>0 \\) on these intervals. Hence the maximum value is \\( \\max \\{f(-2), f(3)\\}=18 \\).",
  "vars": [
    "x"
  ],
  "params": [
    "f"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variable",
        "f": "function"
      },
      "question": "Find, with explanation, the maximum value of $function(variable)=variable^3-3variable$ on the\nset of all real numbers $variable$ satisfying $variable^4+36\\leq 13variable^2$.",
      "solution": "Solution. The condition \\( variable^{4}+36 \\leq 13 variable^{2} \\) is equivalent to \\( (variable-3)(variable-2)(variable+2)(variable+3) \\leq \\) 0 , which is satisfied if and only if \\( variable \\in[-3,-2] \\cup[2,3] \\). The function \\( function \\) is increasing on \\( [-3,-2] \\) and on \\( [2,3] \\), since \\( function^{\\prime}(variable)=3\\left(variable^{2}-1\\right)>0 \\) on these intervals. Hence the maximum value is \\( \\max \\{function(-2), function(3)\\}=18 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "tangerine",
        "f": "porcupine"
      },
      "question": "Find, with explanation, the maximum value of $porcupine(tangerine)=tangerine^3-3tangerine$ on the\nset of all real numbers $tangerine$ satisfying $tangerine^4+36\\leq 13tangerine^2$.",
      "solution": "Solution. The condition \\( tangerine^{4}+36 \\leq 13 tangerine^{2} \\) is equivalent to \\( (tangerine-3)(tangerine-2)(tangerine+2)(tangerine+3) \\leq \\) 0 , which is satisfied if and only if \\( tangerine \\in[-3,-2] \\cup[2,3] \\). The function \\( porcupine \\) is increasing on \\( [-3,-2] \\) and on \\( [2,3] \\), since \\( porcupine^{\\prime}(tangerine)=3\\left(tangerine^{2}-1\\right)>0 \\) on these intervals. Hence the maximum value is \\( \\max \\{porcupine(-2), porcupine(3)\\}=18 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "knownvalue",
        "f": "nonfunction"
      },
      "question": "Find, with explanation, the maximum value of $nonfunction(knownvalue)=knownvalue^3-3knownvalue$ on the\nset of all real numbers $knownvalue$ satisfying $knownvalue^4+36\\leq 13knownvalue^2$.",
      "solution": "Solution. The condition \\( knownvalue^{4}+36 \\leq 13 knownvalue^{2} \\) is equivalent to \\( (knownvalue-3)(knownvalue-2)(knownvalue+2)(knownvalue+3) \\leq \\) 0 , which is satisfied if and only if \\( knownvalue \\in[-3,-2] \\cup[2,3] \\). The function \\( nonfunction \\) is increasing on \\( [-3,-2] \\) and on \\( [2,3] \\), since \\( nonfunction^{\\prime}(knownvalue)=3\\left(knownvalue^{2}-1\\right)>0 \\) on these intervals. Hence the maximum value is \\( \\max \\{nonfunction(-2), nonfunction(3)\\}=18 \\)."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "f": "hjgrksla"
      },
      "question": "Find, with explanation, the maximum value of $hjgrksla(qzxwvtnp)=qzxwvtnp^3-3qzxwvtnp$ on the\nset of all real numbers $qzxwvtnp$ satisfying $qzxwvtnp^4+36\\leq 13qzxwvtnp^2$.",
      "solution": "Solution. The condition \\( qzxwvtnp^{4}+36 \\leq 13 qzxwvtnp^{2} \\) is equivalent to \\( (qzxwvtnp-3)(qzxwvtnp-2)(qzxwvtnp+2)(qzxwvtnp+3) \\leq \\) 0 , which is satisfied if and only if \\( qzxwvtnp \\in[-3,-2] \\cup[2,3] \\). The function \\( hjgrksla \\) is increasing on \\( [-3,-2] \\) and on \\( [2,3] \\), since \\( hjgrksla^{\\prime}(qzxwvtnp)=3\\left(qzxwvtnp^{2}-1\\right)>0 \\) on these intervals. Hence the maximum value is \\( \\max \\{hjgrksla(-2), hjgrksla(3)\\}=18 \\)."
    },
    "kernel_variant": {
      "question": "Determine, with proof, the maximum value of the function\n\\[\n\\displaystyle f(x)=x^{3}-2x\n\\]\nover all real numbers \\(x\\) that satisfy the inequality\n\\[\n x^{4}+25\\;\\le\\;26x^{2}.\n\\]",
      "solution": "First rewrite the constraint in a factored form.  \n\\[  \n x^{4}+25\\le 26x^{2}\\;\\iff\\;x^{4}-26x^{2}+25\\le 0.  \n\\]  \nObserve that  \n\\[  \n x^{4}-26x^{2}+25=(x^{2}-25)(x^{2}-1)=(x-5)(x-1)(x+1)(x+5).  \n\\]  \nHence the inequality is satisfied precisely when the product of the four linear factors is non–positive, that is,  \n\\[  \n x\\in[-5,-1]\\;\\cup\\;[1,5].\\tag{1}  \n\\]\nNext examine the monotonicity of the objective function on each of the two admissible intervals.  \n\\[  \n f'(x)=3x^{2}-2.  \n\\]  \nThe derivative vanishes when \\(3x^{2}-2=0\\), i.e. at \\(x=\\pm\\sqrt{\\tfrac23}\\approx\\pm0.816\\).  These critical points lie strictly between \\(-1\\) and \\(1\\).  Consequently,  \n\\[  \n f'(x)=3x^{2}-2>0\\quad\\text{for}\\quad |x|\\ge1.  \n\\]  \nIn particular, \\(f'\\) is positive on each of the admissible intervals (1).  Therefore \\(f\\) is strictly increasing on both \\([-5,-1]\\) and on \\([1,5]\\).\n\nBecause \\(f\\) is increasing on each interval, its maximum over the entire feasible set occurs at the right–hand endpoint of each interval.  Thus  \n\\[  \n f(-1)=(-1)^{3}-2(-1)=-1+2=1,\\qquad f(5)=5^{3}-2\\cdot5=125-10=115.  \n\\]  \nComparing the two values, the larger is 115.\n\nHence the maximum value of \\(f(x)=x^{3}-2x\\) subject to \\(x^{4}+25\\le26x^{2}\\) is  \n\\[  \n\\boxed{115}.  \n\\]",
      "_meta": {
        "core_steps": [
          "Rewrite the constraint as a factored quartic (difference-of-squares) to get two closed intervals for x",
          "Differentiate f and check the sign of f′ on each admissible interval to see that f is increasing there",
          "Conclude that the maximum of f occurs at the right-hand endpoint of each interval and evaluate f at those endpoints"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "The two positive numbers whose squares give the four real roots of the quartic, hence the endpoints of the admissible intervals",
            "original": "3 and 2 (from (x−3)(x−2)(x+2)(x+3)≤0 → intervals [−3,−2]∪[2,3])"
          },
          "slot2": {
            "description": "The coefficient of x in f(x)=x³−c x; it controls where f′(x)=3x²−c vanishes, which must be inside (−β,β) so that f′ keeps one sign on each admissible interval",
            "original": "c = 3  (critical points at ±1 < 2)"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}