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Question

Which of the following statements is/are correct in stating the number of solutions of the equation
xlogx+x=λ, where λR.

A
If λ=0, then the equation has exactly two solutions in x.
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B
If λ(0,1e2), then the equation has exactly one solution in x.
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C
If λ(1e2,0), then equation has at least one solution in x.
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D
the equation has at least one solution in [1,λ]
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Solution

The correct option is D the equation has at least one solution in [1,λ]
x logx+x=λ(i)
Let f(x)=xlogx+x=0xlogx+x=0x(logx+1)=0
x=0 or x=1e
x=0 is not a solution as log of 0 is not defined.
So, the equation has exactly one solution if λ=0

(ii) Exactly 1 solution in x if λ(0,1e2)
(iii) At least 1 solution in x if λ(1e2,0)
limx0+f(x)=limx0+(xlogx+x)=limx0+logx+11x
Using L'hospitals' rule, we get
limx0+1x1x2=limx0+(x)=0

f(x)=x×1x+logx+1=2+logxf(x)=0logx=2x=1e2
f′′(x)=1x>0 at x=1e2
Hence, x=1e2 is a point of local minima.

f(1)=2 (>0)
f(1e3)=1 (<0)

If λ>0 Only 1 solution
If λ<0 2 solutions.

(iv) At least one solution in [1,λ]
f(x)=xlogx+x=x(1+logx)f(x)x, x[1,λ]

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