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Question

At which of the following points does f(x)=x4 has a maximum?


A

0

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B

1

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C

2

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D

3

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E

None of these

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Solution

The correct option is E

None of these


We saw that the necessary condition for f(x) to have a maximum at x=cisf(c)=0

We have f(x)=4x3

f(x)=0x=0. Now we have to check if x=0 is a local maxima. For that, we have to check with nth derivative test conditions -

If f(x) has derivative upto nth order and f(c)=f(c)..fn1(c)=0, then

A) n is even, fn(c)<0x=c is a point of maximum

B) n is even, fn(c)>0x=c is a point of minimum

C) n is odd, fn(c)<0f(x) is decreasing about x=c

D) n is odd, fn(c)>0f(x) is increasing about x=c

So, we will differentiate the given function until we get a non negative value at x=0

f(x)=x4

f(x)=4x3

f(x)=12x2

f(0)=0

f(x)=f3(x)=24x

f3(0)=0

f(x)=f4(x)=24

f4(x)=24

n is even, fn(c)>0x=c is a point of minimum

x=0 is a local minimum. So this function does not have any local maximum.


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