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Question

A cubic polynomial f(x) vanishes at x=2 and has a relative minimum/maximum at x=1 and x=13. Then

A
If 11f(x) dx=133, then f(x)=14(5x3+5x25x+5)
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B
If 11f(x) dx=143, then f(x)=x3+x2x+2
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C
If 11f(x) dx=53, then f(x)=16(5x3+5x25x+15)
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D
If 11f(x) dx=103, then f(x)=17(5x3+5x25x+10)
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Solution

The correct options are
B If 11f(x) dx=143, then f(x)=x3+x2x+2
D If 11f(x) dx=103, then f(x)=17(5x3+5x25x+10)
As the function has relative minimum/maximum at x=1 and x=13
So, f(x)=a(x+1)(x13)
where a is constant.
f(x)=a(x2+2x313)
f(x)=a(x33+x23x3)+b
where b is constant of integration.

Now f(2)=0
8a3+4a3+2a3+b=0
b=2a3
f(x)=a3(x3+x2x+2)

When 11f(x) dx=143
a311(x3+x2x+2) dx=143
a11(x2+2)=14
2a10(x2+2)=14
2a(13+2)=14
143a=14a=3

So the function will be
f(x)=x3+x2x+2

Similarly, when 11f(x) dx=103
143a=10
a=157

So the function will be
f(x)=17(5x3+5x25x+10)
Similarly, remaining two options can be found to be incorrect.

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