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Question

Let f be real valued function on R defined as f(x)=x4(1x)2. Then which of the following statements is (are) CORRECT?

A
f(x)=0 for some x(0,1)
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B
f′′(x) vanishes exactly twice in R
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C
f(x) is an even function
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D
f(x) is monotonically increasing in (0,23)(1,)
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Solution

The correct options are
A f(x)=0 for some x(0,1)
D f(x) is monotonically increasing in (0,23)(1,)
Here, we can see that f(x)f(x)
So, clearly f(x) is not an even function.

Consider f(x)=x4(1x)2, x[0,1]
f(x) being a polynomial, is continuous and differentiable also.
f(0)=0=f(1)
So, Rolle's theorem is applicable here.
f(x)=4x3+6x510x4
So, there exists at least one x(0,1) such that f(x)=0

f(x)=4x3+6x510x4
=2x3(2+3x25x)
=2x3(x1)(3x2)
f(x)>0 in (0,23)(1,)
So, f(x) is monotonically increasing in (0,23)(1,)


f′′(x)=0
12x2+30x440x3=0
2x2(6+15x220x)=0
x=0 or 6+15x220x=0
15x220x+6=0
D=400360=40>0
So, 15x220x+6=0 has two distinct real roots.
Hence, f′′(x)=0 has three distinct solutions.




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