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Question

The function fx=-1xtet-1t-1t-23t-35dt for t-1,x has a local minimum at,


A

0, 4

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B

1, 3

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C

0, 2

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D

2, 4

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Solution

The correct option is B

1, 3


Explanation for the correct option

Step 1: Solve for the critical points

Given function is,
fx=-1xtet-1t-1t-23t-35dt

Then,
f'x=xex-1x-1x-23x-35ddxaxftdt=fx

The critical points of the functions can be obtained by setting f'x=0
xex-1x-1x-23x-35=0x=0ex-1=0x-1=0x-23=0x-35=0x=0x=1x=2x=3

Step 2: Find the local minima

A function has local minima at a if f'x<0 to its left and f'x>0 to its right.

We see that,
f'-1=-11e-1-1-1-1-23-1-35=-10.3676-1-2-33-45<0e=2.72
And,
f'12=12e-112-112-2312-35=12e-1-12-323-525<0

Thus, fx does not have local minimum at x=0

Also,
f'32=32e32-132-132-2332-35=32e32-112-123-325>0

Thus, fx has local minimum at x=1

f'52=52e52-152-152-2352-35=52e52-132123-125<0
And,
f'72=72e72-172-172-2372-35=72e72-152323125>0

Thus, fx has local minimum at x=3

So, fxhas local minima at x=1 and x=3

Hence option(B) i.e. 1, 3 is correct.


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