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Question

# Let f(x)=⎧⎪ ⎪⎨⎪ ⎪⎩x3+x2−10x;−1≤x<0sin x;0≤x<π2,1+cos x;π2≤x≤π 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 = -1The 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+2x−10;−1≤x<0cos x;0<x<π2−sin x;π2<x≤π 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 −1≤x<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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