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Question

The function f(x)=x1t(et1)(t1)(t2)3(t3)5dt

has a local minimum at x=

A
4
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B
1
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C
0
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D
2
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Solution

The correct option is B 1
We have f(x)=x1t(et1)(t1)(t2)3(t3)5dt

f(x)=x(ex1)(x1)(x2)3(x3)5

For extreme points f(x)=0x=0,1,2,3.

At x=0,f(x) does not change sign as x crosses 0.

At x=1,f(x)>0or<0 according as x>1 or x<1.

f(x) has local minimum at x=1.

At x=2,f(x)>0 or <0 according as x<2 or x>2.

f(x) has local maximum at x=2.

At x=3,f(x)>0 or <0 according as x>3 or x<3.

f(x) has local minima at x=3.

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