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Question

f(x) is cubic polynomial with f(2)=18 and f(1)=1. Also f(x) has local maxima at x=1 and f'(x) has local minima at x=0, then

A
the distance between (1,2) and (a,f(a)), where x=a is the point of local minima is 25
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B
f(x) is increasing for x[1,25]
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C
f(x) has local minima at x=1
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D
the value of f(0)=15
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Solution

The correct option is C f(x) has local minima at x=1
Let f(x)=ax3+bx2+cx+d
f(2)=188a+4b+2c+d=18(1)

f(1)=1a+b+c+d=1(2)

f(x) has local maxima at x=1,
f(1)=03a2b+c=0(3)

f(x) has local minima at x=0,
f′′(0)=0b=0(4)

Solving (1),(2),(3) and (4), we get
f(x)=14(19x357x+34)
or f(0)=172

f(x)=574(x21)>0 x>1
f(x)=0x=1,1
f′′(1)<0,f′′(1)>0
Thus, x=1 is a point of local maximum and x=1 is a point of local minimum.

So a=1,
(a,f(a))=(1,f(1))=(1,1)
The distance between (1,2) and (1,f(1)), i.e., (1,1) is
=22+32=1325

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