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Question

Let a,bR be such that the function f is given by f(x)=ln|x|+bx2+ax,x0 has extreme values at x=1 and x=2.

Statement 1:f has local maximum at x=1 and at x=2.
Statement 2:a=12 and b=14.

A
Statement 1 is false, statement-2 is true.
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B
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
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C
Statement 1 is true, Statement 2 is true; statement 2 is not a correct explanation for statement 1.
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D
Statement 1 is true, statement 2 is false.
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Solution

The correct option is B Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
We have,
f(x)=ln|x|+bx2+ax
f(x)=1x+2bx+a

Now at x=1,f(x)=0
12b+a=0
a2b=1(1)

Also at x=2,f(x)=0
12+4b+a=0
a+4b=12(2)

Solving (1) and (2) we get,
b=14 and a=12

f(x)=1xx2+12
f′′(x)=1x212

Now, f′′(1)=32<0
and f′′(2)=34<0

Hence, f has local maximum at x=1 and at x=2.

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.

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