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Question

Which of the following statement is true for the function
f(x)=xx1=x30x1=x334x x<0

A
It is monotonic increasing for all xϵR
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B
f(x) fails to exist for 3 distinct real values of x
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C
f(x) changes its sign twice as x varies from (,)
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D
function attains its extreme values at x1 & x2, such that x1,x2>0
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Solution

The correct option is D f(x) changes its sign twice as x varies from (,)
For x<0
f(x)=x334x
Hence
f(x)=x24
Now for monotonically increasing function
f(x)>0
Or
xϵ(,2)(2,). However domain is x<0.
Hence xϵ(.2).
For
0<x<1
f(x)=x3.
f(x)=3x2
Now for monotonically increasing function f(x)>0
Or
x>0.
Hence f(x) is an increasing function in (0,1).
For x>1
f(x)=x
f(x)=122
for monotonically increasing function f(x)>0
Hence
x>0.
Thus summing up, we get f(x) as an increasing function in the interval of
(,2)(0,).
Hence it changes sign twice, once at x=2 and another at x=0.

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