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Question

If g(x)=k has exactly two solutions then k belong to

A
(2,1)
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B
(1,0)
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C
(3,2)
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D
None of these
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Solution

The correct option is D None of these
Let f(x)=u(x)2v(x)
where u(x)=min(x2,|x|);|x|1max(x2,|x|);|x|>1
from the figure
Clearly u(x)=x2
v(x)=min(x2,|x|);|x|>1max(x2,|x|);|x|1
Clearly, v(x)=|x|
and g(x)=minf(t);3tx;3x0maxf(t);0tx;0x3
f(x)=u(x)2v(x)=x22|x|
g(x)=min(f(t))=t22|t|=t2+2t, if 3<t<x
g(x)=x2+2x if x < -1, else g(x)=1 for x>-1
g(x)=max(f(t))=t22|t|=t22t, if 0<t<x
g(x)=0 if x<2, else g(x)=x22x for x>2
352356_130202_ans.png

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