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Question

Let f(x)=⎪ ⎪⎪ ⎪x3+x210x;1x<0sin x;0x<π2,1+cos x;π2xπ then f(x) has

A
local maxima at x = π2
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B
local minima at x = π2
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C
absolute maxima at x = -1
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D
absolute maxima at x=π2
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Solution

The correct options are
A local maxima at x = π2
C absolute maxima at x = -1
The function f is not differentiable at x = 0,

x=π2 as f(0)=10,f(0+)=1;

f(π2)=0,f(π2+)=1.The function

f(x) is given by

f(x)=⎪ ⎪⎪ ⎪3x2+2x10;1x<0cos x;0<x<π2sin x;π2<xπ

The critical points of f are given by

f(x)=0 or x=0,π2,π.

Since f(x)>0, for 0<x<π2 and f(x)<0, for

π2 < x < π so f has local maxima at
x=π2.

Also f(x)<0 for 1x<0 and f(x)>0 for
0<x<π2 so f has local minima at x = 0.

Since f(1)=10,f(π2)=1,f(0)=0 and f(π)=0.

Thus f has absolute maximum at x=1 and absolute minimum at x=0

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