f(x)=cosx/x是否有界，要步骤 函数f(x)=xcosx在(一∞，十∞)上有界
xcosx\u662f\u4e0d\u662f\u6709\u754c\u51fd\u6570\uff0c\u6c42\u8fc7\u7a0b\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002\u3002
\u8bbef\uff08x\uff09=xcosx\uff0c\u5982\u679cf\uff08x\uff09\u5b58\u5728\u4e0a\u754ct>0\u3002\u4f7f\u5f97x\u2208R\u65f6\uff0c\u6709f(x)\u2264t,
\u90a3\u4e48\u53d6x=2\u03c0t\uff0cf\uff08x)=xcosx=2\u03c0t>t\uff0c\u77db\u76fe\uff0c\u6545f\uff08x\uff09\u4e0d\u5b58\u5728\u4e0a\u754c\u3002
\u4e0b\u754c\u7c7b\u4f3c\u3002
\u89e3\uff1af(x\uff09=xcosx\u3002
\u662fu=x\u548cv=cosx\u7684\u79ef\u51fd\u6570\u3002
f=uv\u3002
\u5b9a\u4e49\u57df\u4e3au\u548cv\u7684\u5b9a\u4e49\u57df\u7684\u4ea4\u96c6\u3002
u=x\u7684\u5b9a\u4e49\u57df\u4e3aR,v=cosx\u7684\u5b9a\u4e49\u57df\u4e3aR
R\u4ea4R=R\u3002
R\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\uff0c
\u5728R\u4e2d\u4efb\u53d6\u4e00\u70b9x.
f(-x)=(-x)cos(-x)=-xcosx=-f(x)
f(-x)+f(x)=0\u5bf9\u4e8ex:R\u4e0a\u6052\u6210\u7acb\u3002
\u6240\u4ee5f(x)\u662f\u5947\u51fd\u6570\u3002
\u56fe\u50cf\u5173\u4e8e\u539f\u70b9\uff080\uff0c0\uff09\u4e2d\u5fc3\u5bf9\u79f0\uff0c
\u5148\u753b\u51fa\u5728\u534a\u533a\u95f4[0,+\u65e0\u7a77\uff09\u4e0a\u7684\u56fe\u50cf\uff0c\u7136\u540e\u518d\u628a\u56fe\u50cf\u5173\u4e8e\uff080,0\uff09\u987a\u65f6\u9488\u65cb\u8f6c180\u5ea6\uff0c\u5373\u5f97\u5bf9\u57ce\u533a\u95f4\uff08-\u65e0\u7a77\uff0c0]\u4e0a\u7684\u56fe\u50cf\uff0c\u4e24\u4e2a\u533a\u95f4\u7684\u4ea4\u96c6\u4e3a{0},\u4e8c\u8005\u6709\u516c\u5171\u90e8\u5206\uff0c\u516c\u5171\u90e8\u5206\u5c31\u4e3ax=0\u8fd9\u4e2a\u70b9\uff0c\u7136\u540e\u5e76\u96c6\uff08-\u65e0\u7a77\uff0c0]u[0,+\u65e0\u7a77\uff09=\uff08-\u65e0\u7a77\uff0c0\uff09u{0}u[0,+\u65e0\u7a77\uff09=\uff08-\u65e0\u7a77\uff0c0\uff09u[0,+\u65e0\u7a77\uff09u{0}=Ru{0}=R\u3002
\u6240\u4ee5\u5728R\u4e0a\u7684\u56fe\u50cf\u5c31\u5168\u90e8\u753b\u4e86\u51fa\u6765\u3002
f(x)=xcosx\u3002
/f(x)/=/xcosx/=/x//cosx/,
\u5bf9\u4e8ex:R,cosx\u5c5e\u4e8e[-1,1]
x:[0,+\u65e0\u7a77\uff09\u771f\u5305\u542b\u4e8eR,\u662fR\u7684\u5b50\u533a\u95f4\uff0c\u5728R\u4e0a\u6210\u7acb\uff0c\u5728[0,+\u65e0\u7a77\uff09\u4e0a\u4e00\u5b9a\u6210\u7acb
\u5219cosx:[-1,1]
0<=/cosx/<=1
0<=/cosx/<=1.
\u5f53/cosx/=0\u65f6\uff0ccosx=0,x=kpai,k\uff1aZ\u3002x\u4e3a\u7ec8\u8fb9\u5728x\u8f74\u4e0a\u7684\u8f74\u5411\u89d2\uff0c
\u5219f\uff08x)=x*0=0\u3002
\u5f53x=kpai,k:Z,
x>0
kpai>0
k>0,k:Z
k:Z+
k=1,2,3.......
x=pai,2pai,3pai,.........kpai,.....(k:Z+)
\u5728\u8fd9\u4e9b\u70b9\u4e0af(x)=0\u3002
2.\u5f53/cosx/=0\u65f6\uff0c\u5373\u628a/cosx/=0\u4ece[0,1]\u4e2d\u53bb\u9664\u6389\uff0c\u53730</cosx/<=1
x>=0
\u5f53x=0\u65f6\uff0cf(x)=0\u3002
\u5f53x>0\u65f6\uff0c/x/=x>0\u3002
0</cosx/<=1
/x/>0
\u5219\u4e0d\u7b49\u5f0f\u4e24\u8fb9\u7edf\u79f0\u4ee5/x/,\u4e0d\u7b49\u53f7\u4fdd\u6301\u4e0d\u53d8\u3002
0</cosx//x/<=/x/
0</xcosx/<=x\u3002
0</f(x)/<=x\u3002
-x<=f(x)<0or0<f(x)<=x\u3002
\u753b\u51fay=-x\u548cy=x\u7684\u56fe\u50cf\u5728[0,+\u65e0\u7a77\uff09\u4e0a\u7684\u56fe\u50cf\uff0c\u4e3a\u901a\u8fc7\u539f\u70b9\u7684\u4e24\u6761\u5173\u4e8ex\u8f74\u5bf9\u79f0\u7684\u5c04\u7ebf\u3002
\u56e0\u4e3a{0}u[-x,0)u(0,x)=[-x,0]u(0,x]=[-x,x],x>=0\u3002
f(x)\u5728-x\u548cx\u4e4b\u95f4\uff0c\u4e0d\u4f1a\u8d85\u8fc7\u8fd9\u4e24\u4e2a\u503c
\u90a3\u4e48f(x)\u7684\u56fe\u50cf\u4e00\u5b9a\u5728y=-x\u548cy=x\u7684\u56fe\u50cf\u4e4b\u95f4\uff0c\u4e0d\u4f1a\u8d85\u51fa\uff0c\u6700\u591a\u548c\u8fd9\u4e24\u6761\u5c04\u7ebf\u76f8\u5207\uff0c
\u662f\u6b63\u5f53\u7684\u66f2\u7ebf\u3002
\u7136\u540e\u4f1a\u901a\u8fc7\uff080,0),(pai,0\uff09.......(kpai,0),k:Z+
\u5373\u4e0ex\u8f74\u7684\u4ea4\u70b9\u3002
\u7136\u540e\u5728[0,pai]\u4e0a\u662f\u5728x\u8f74\u4e0a\u65b9\uff0c\u5728\uff08pai,2pai]\u5728x\u8f74\u7684\u4e0b\u65b9\uff0c\u4ea4\u66ff\u51fa\u73b0\uff0c
\u7136\u540e\u5728[(k-1)pai,kpai]\u5185\u7684\u7edd\u5bf9\u503c\u6700\u503c\u968f\u7740k\u7684\u589e\u5927\u800c\u589e\u5927\uff0c
\u5373\u9707\u8361\u7684\u5e45\u5ea6\u9010\u6e10\u589e\u5927\uff0ck-\u65e0\u7a77\uff0c\u5219\u6b63\u8d1f-\u65e0\u7a77\u5927\u3002
\u540c\u7406\uff0c\u5728\uff08-\u65e0\u7a77\uff0c0]\u4e0a\u7684\u56fe\u50cf\u5173\u4e8e\uff080,0)\u5bf9\u79f0\uff0c\u4e5f\u5728y=-x\u548cy=x\u4e4b\u95f4\u4e89\u5f53\u3002
\u5728R\u8fd9\u4e2a\u65e0\u7a77\u533a\u95f4\u4e0a\u65e0\u9650\u5730\u6b63\u5f53\u4e0b\u53bb\uff0c\u8fd9\u4e2a\u56fe\u50cf\u5728[0,+\u65e0\u7a77\uff09\u968f\u7740x\u7684\u589e\u5927\uff0c\u5219\u9707\u8361\u8d8a\u6765\u8d8a\u5267\u70c8\uff0c
k\u589e\u5927\uff0c\u5219\u5728\u533a\u95f4[\uff08k-1)\u03c0\uff0ckpai)\u5185,k:Z+,f(x)\u7684\u6700\u503c\u5f97\u7edd\u5bf9\u503c\u968fk\u7684\u589e\u5927\u800c\u589e\u5927\uff0c\u8bbe\u5728\u8fd9\u4e4b\u95f4\u7684/f(x)/\u7684\u6700\u503c=f(x0),x0\u5c5e\u4e8e[\uff08k-1)\u03c0\uff0ckpai)\u5185,k:Z+\uff0c\u968f\u7740x\u7684\u589e\u5927\uff0c\u7136\u540e/f(x)/max\u589e\u5927\uff0c\u56e0\u4e3ax-+\u65e0\u7a77\uff0c\u5219/f(x)/max-+\u65e0\u7a77\uff0c\u56e0\u4e3a/f(x)/max\u5728\uff080\uff0c+\u65e0\u7a77\uff09\u4e0a\u662f\u5355\u8c03\u9012\u589e\u7684\uff0c\u6240\u4ee5x-\u65e0\u7a77\u5927\uff0c/f(x)/max-\u65e0\u7a77\u5927\uff0cf(x)>0\u65f6\uff0cf(x)max-\u65e0\u7a77\u5927\uff0cf(x)max\u4e0d\u5b58\u5728\uff0c\u5f53f(x)<0\u65f6\uff0c\uff08-f(x))max-\u65e0\u7a77\u5927\uff0cf(x)min-\u65e0\u7a77\u5c0f\uff0c\u5373\u6781\u5c0f\u503c-\u65e0\u7a77\u5c0f\uff0c\u65e0\u7a77\u5c0f\u65f6\u4e0d\u5b58\u5728\uff0c\u6240\u4ee5f(x)\u7684\u6700\u5c0f\u503c\u4e0d\u5b58\u5728\uff0cf(x)\u65e2\u6ca1\u6709\u6700\u5927\u503c\uff0c\u4e5f\u6ca1\u6709\u6700\u5c0f\u503c\uff0cf(x)\u662f\u65e0\u754c\u51fd\u6570
\u6839\u636e\u5bf9\u79f0\u6027\uff0c\u5728\uff08-\u65e0\u7a77\uff0c0]\u4e0af(x)\u4e5f\u662f\u65e0\u754c\u51fd\u6570
\u7efc\u4e0af(x)\u5728\uff08-\u65e0\u7a77\uff0c+\u65e0\u7a77\uff09\u4e0a\u662f\u65e0\u754c\u51fd\u6570\u3002
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