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Question Number 100032 by bobhans last updated on 24/Jun/20
Givenf(xx+1)=x2.findminimumvalueoffunctionh(x)=f(x)−3x−1
Commented by john santu last updated on 24/Jun/20
letxx+1=z⇒xz+z=xx=z1−z⇒f(z)=z2(1−z)2similartof(x)=x2(1−x)2.h(x)=x2(1−x)2+31−xh(x)=x2−3x+3(1−x)2h′(x)=(2x−3)(1−x)2+2(x2−3x+3)(1−x)(1−x)4=0⇒(1−x){(2x−3)(1−x)+2x2−6x+6}=0(1−x){−2x2+5x−3+2x2−6x+6}=0(1−x)(3−x)=0,criticalpointx=3minh(3)=34
Answered by mathmax by abdo last updated on 24/Jun/20
letxx+1=t⇒x=tx+t⇒(1−t)x=t⇒x=t1−t⇒f(t)=t2(1−t)2⇒f(x)=x2(x−1)2=x2x2−2x+1⇒g(x)=x2(x−1)2−3x−1=x2−3(x−1)(x−1)2=x2−3x+3(x−1)2gdefinedonR{1}limx→∞g(x)=1andlimx→1−g(x)=+∞=limx→1+g(x)g′(x)=(2x−3)(x−1)2−2(x−1)(x2−3x+3)(x−1)4=′(2x−3)(x−1)−2(x2−3x+3)(x−1)3=2x2−5x+3−2x2+6x−6(x−1)3=(x−3)(x−1)(x−1)4andg′(x)=0⇔x=3x−∞13+∞g′(x)+∣∣−0+g(x)1inc+∞+∞decrg(3)inc1g(3)=34istheminimumvalueforg
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