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Question Number 106309 by bemath last updated on 04/Aug/20
Ifg(x)=x+xandlimx→2f(x)−f(2)x2+ax+b=43findthevalueof(f∘g)′(1).
Answered by bobhans last updated on 05/Aug/20
g(x)=x+x⇒g′(x)=1+12x;g′(1)=32g(1)=2(f∘g)′(1)=g′(1)f′(g(1))=32f′(2)(1)22+2a+b=0→b=−4−2a(2)limx→2f(x)−f(2)x2+ax−2a−4=43limx→2f′(x)2x+a=43case(1)ifa≠−4,thenwegetf′(2)=16+4a3.so(f∘g)′(1)=g′(1).f′(g(1))=32.f′(2)=32.16+4a3=8+2acase(2)ifa=−4,thenlimx→2f′(x)2x+aisform00sof′(2)=0⇒(f∘g)′(1)=32.0=0
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