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Question

Let the function be defined as follows: f(x)=x3+x210x,1x<0
cosx,0x<π2
1+sinx,π2xπ. Then f(x) has

A
a local minimum at x=π/2
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B
a local maximum at x=π/2
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C
a local minimum at x=1
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D
a local maximum at x=π
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Solution

The correct option is B a local maximum at x=π/2
The function f(x) is given by
f(x)=3x2+2x10-1x<0sinx0x<π/2cosx\pi/2xπ
The function f(x) is not differentiable at x=0, x=π/2
as f(0)=10, f(0+)=0; f(π/2)=1, f(π/2+)=0.
The critical points of f are given by f(x)=0 or x=0, π/2.
Since f(x)<0 for 1x0 and f(x)<0 for 0x<π/2
Therefore, f(x) does not have any extremum at x=0
Also f(x)<0 for 0x<π/2 and f(x)<0 for π/2xπ
Therefore, f(x) does not have any extremum at x=π/2

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