多元变量最值问题解法探析-方立新

20158、、。、。、。。、。、。1abc∈0+∞3a-2b+c=0ac姨b________。abc∈0+∞2b=3a+c≥23ac姨ac姨b≤13姨。2xyzx2-3xy+4y2-z=0xyz2x+1y-2z_________。x2-3xy+4y2-z=0z=x2-3xy+4y2xyz=xyx2-3xy+4y2=1xy+4yx-3≤124姨-3=1xy=4yxx=2y。z=2y22x+1y-2z=-1y2+2y=-1y-2212+1y=12x+1y-2z22max=1。【】。、。。3abab-4a-b+1=0a0a+1b+2_______。ab-4a-b+1=0b=4a-1a-1。a+1b+2=a+14a-1a-1+222。t=a-1t+23t+226=15+6t+6t≥27。【】。。4xfx=ax2+bx+c≥0aba+b+cb-a_________。fx=ax2+bx+c≥0a0b2-4ac≤0c≥b24a。ba=t1a+b+cb-a≥a+b+b24ab-a=4+4t+t24t-1=14t-1+9t-1+226≥3。、。5xy4x2+y2+xy=12x+y__________。4x2+y2+xy=12x+y2-3xy=1。2x+y=t2x·y≤2x+y2222xy≤t28t2-1≤38t2。t2≤85-210姨5≤t≤210姨5。2x+y210姨5。【】2x+y4320158。6xyx3x+y+yx+2y________。3x+y=mx+2y=nx=2m-n5y=3n-m5。x3x+y+yx+2y=2m-n5m+3n-m5n=1-n5m-m5n≤1-25=35。x3x+y+yx+2y35。【】xy。、7abcabc+a+c=bp=2a2+1-2b2+1+3c2+1p。a=tanαb=tanβc=tanγαβγ∈0π2∈∈p=2cos2α-2cos2β+3cos2γ=cos2α-cos2β+3cos2γ=2sinα+βsinβ-α+3cos2γ。abc+a+c=btanβ=b=a+c1-ac=tanα+tanγ1-tanαtanγ=tanα+γ。αβγ∈0π2ttβ=α+γγ=β-α。p=2sinα+βsinγ+3cos2γ≤2sinγ+3cos2γ=-3sinγ-13tt2+103≤103。p103。【】p。8ab∈Ra2+2b2=6ba-3_________。a=6姨cosθb=3姨sinθt=ba-3=3姨sinθ6姨cosθ-3=sinθ2姨cosθ-3姨t2姨cosθ-3姨=sinθ2姨tcosθ-sinθ=3姨t。2t2+1姨cosθ+φ=3姨ttanφ=12姨t。cosθ+φ=3姨t2t2+1姨cosθ+φ=3姨t2t2+1姨≤1t2≤1。t1。【】。ab30。、9△ABCabcb+c≤2ac+a≤2b△ba_______。abca+bca+cbb+ca△△△△△△△△△aba+ca≤2ca+1≤2ba△△△△△△△△△△△1+baca1+cababa+ca1△△△△△△△△△△△△△△△△△ba=xca=yx+y≤2y+1≤2x1+xy1+yxx+y1△△△△△△△△△△△△△△△△△y≤-x+2yx+1yx-1y-x+1y≤2x-1△△△△△△△△△△△△△△△△△x2332∈tba2332∈t。【】abc。、10Axyy=1xx3+y3≥ax+y2a____。xy=1x0y0x3+y3≥ax+y2a≤x3+y3x+y2。x3+y3x+y2=x+yx2-xy+y2x+y2=x2-xy+y2x+y=x+y-3xyx+y=x+y-3x+yt=x+y≥2xy姨=2t-3t2＋∞t=2t-3tttmin=12。a≤12。【】x+yt。11abcc1a2+tb2-4c2≥0t___________。4420158tb2≥4c2-1a2t≥b24c2-1a222b2=c2-a2t≥c2-a24c2-1a222。c2-a24c2-1a222=5-4a2c2+c2a222≤1。tt≥1。【】。、12a2+8b2≥γba+bab∈Rγ________。1aa2-γab+8-γb2≥0a。Δ=-γb2-48-γb2≤0γ2－48－γb2≤0bγ2-48-γ≤0-8≤γ≤4。2b=0γ∈Rb≠0b2ab222+8≥γab+221。t=abt2-γt+8-γ≥0Δ≤0-8≤γ≤4。γ－84。【】。。。1。。。。1。2。1。2。3。……11.、。、。。。2.、———《》≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠45