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Question

# If f(x)=alog|x|+bx2+x has its extremum values at x=âˆ’1 and x=2, then

A
a=2,b=1
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B
a=2,b=12
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C
a=2,b=12
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D
None of the above.
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Solution

## The correct option is B a=2,b=−12The given information in the question is:f(x)=alog|x|+bx2+xAn extremum is calculated from the derivative of the function about a point where the derivative is equal to 0.So we calculate the derivative and equate it to 0.⇒f′(x)=a|x|x×1|x|+2bx+1⇒f′(x)=ax+2bx+1Now equating f′(x) to zero.⇒f′(x)=0⇒ax+2bx+1=0Now it is given that the extremum is at x=−1 andx=2So substituting the value of extremum in the derivative equation which is its solution, we get⇒−a−2b+1=0 .....(I)⇒a2+4b+1=0 .....(II)Solving equations (I) and (II) we get,⇒a+2b=1 .....(III)⇒a+8b=−2 .....(IV)Subtracting (Iv) from (III) we get,⇒b=−12Substituting the value of b In equation (III) we get,⇒a=2For the function f(x) to have an the extremum at the required point the value of a=2. and b=−12. .....Answer[Option(B)]

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