![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag28-3.png\" "\\"Gr\u00e1fico\\"")
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\nO per\u00edodo positivo m\u00ednimo das fun\u00e7\u00f5es $f$e $g$ \u00e9 $2\\pi $.
\nO per\u00edodo positivo m\u00ednimo da fun\u00e7\u00e3o $h$ \u00e9 $\\frac{{2\\pi }}{3}$, pois $3x = 2\\pi\u00a0 \\Leftrightarrow x = \\frac{{2\\pi }}{3}$.<\/p>\nTendo em conta o dom\u00ednio considerado para cada uma das fun\u00e7\u00f5es, conclui-se que qualquer uma n\u00e3o \u00e9 fun\u00e7\u00e3o par, nem fun\u00e7\u00e3o \u00edmpar, pois qualquer dos gr\u00e1ficos n\u00e3o \u00e9 sim\u00e9trico relativamente ao eixo das ordenadas, nem sim\u00e9trico relativamente \u00e0 origem do referencial.<\/p>\n
Quanto aos contradom\u00ednios, $D{‘_f} = D{‘_h} = \\left[ { – 1,1} \\right]$ e $D{‘_g} = \\left[ { – 3,3} \\right]$.
\n\u00ad<\/li>\n
Os gr\u00e1ficos de $f$ e $h$ intersetam-se nos pontos de coordenadas: $$\\begin{array}{*{20}{l}}
\n{\\left( { – \\frac{\\pi }{2},0} \\right)}&,&{\\left( {0,1} \\right)}&,&{\\left( {\\frac{\\pi }{2},0} \\right)}&,&{\\left( {\\pi , – 1} \\right)}&{\\text{e}}&{\\left( {\\frac{{3\\pi }}{2},0} \\right)}
\n\\end{array}$$
\ncujas abcissas se podem confirmar analiticamente:
\n$$\\begin{array}{*{20}{l}}
\n{f(x) = h(x)}& \\Leftrightarrow &{\\begin{array}{*{20}{l}}
\n{\\cos x = \\cos (3x)}& \\wedge &{x \\in \\left[ { – \\frac{\\pi }{2},\\frac{{3\\pi }}{2}} \\right]}
\n\\end{array}} \\\\
\n{}& \\Leftrightarrow &{\\begin{array}{*{20}{l}}
\n{3x =\u00a0 \\pm x + 2k\\pi ,k \\in \\mathbb{Z}}& \\wedge &{x \\in \\left[ { – \\frac{\\pi }{2},\\frac{{3\\pi }}{2}} \\right]}
\n\\end{array}} \\\\
\n{}& \\Leftrightarrow &{\\begin{array}{*{20}{l}}
\n{\\left( {\\begin{array}{*{20}{c}}
\n{2x = 2k\\pi }& \\vee &{4x = }
\n\\end{array}2k\\pi ,k \\in \\mathbb{Z}} \\right)}& \\wedge &{x \\in \\left[ { – \\frac{\\pi }{2},\\frac{{3\\pi }}{2}} \\right]}
\n\\end{array}} \\\\
\n{}& \\Leftrightarrow &{\\begin{array}{*{20}{l}}
\n{\\left( {\\begin{array}{*{20}{c}}
\n{x = k\\pi }& \\vee &{x = }
\n\\end{array}\\frac{{k\\pi }}{2},k \\in \\mathbb{Z}} \\right)}& \\wedge &{x \\in \\left[ { – \\frac{\\pi }{2},\\frac{{3\\pi }}{2}} \\right]}
\n\\end{array}} \\\\
\n{}& \\Leftrightarrow &{\\begin{array}{*{20}{l}}
\n{x = \\frac{{k\\pi }}{2},k \\in \\mathbb{Z}}& \\wedge &{x \\in \\left[ { – \\frac{\\pi }{2},\\frac{{3\\pi }}{2}} \\right]}
\n\\end{array}} \\\\
\n{}& \\Leftrightarrow &{x \\in \\left\\{ { – \\frac{\\pi }{2},0,\\frac{\\pi }{2},\\pi ,\\frac{{3\\pi }}{2}} \\right\\}}
\n\\end{array}$$
\n\u00ad<\/li>\nPor interpreta\u00e7\u00e3o do gr\u00e1fico, tem-se:
\n$$\\begin{array}{*{20}{c}}
\n{f(x) \\geqslant h(x)}& \\Leftrightarrow &{x \\in \\left[ { – \\frac{\\pi }{2},\\frac{\\pi }{2}} \\right]}
\n\\end{array} \\cup \\left\\{ {\\pi ,\\frac{{3\\pi }}{2}} \\right\\}$$
\n$$\\begin{array}{*{20}{c}}
\n{\\frac{{f(x)}}{{h(x)}} \\geqslant 0}& \\Leftrightarrow &{x \\in \\left] { – \\frac{\\pi }{6},\\frac{\\pi }{6}} \\right[}
\n\\end{array} \\cup \\left] { – \\frac{{5\\pi }}{6},\\frac{{7\\pi }}{6}} \\right[$$<\/li>\n<\/ol>\n