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Question

# Let the function be defined as follows: f(x)=x3+x2−10x,−1≤x<0cosx,0≤x<π21+sinx,π2≤x≤π. 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/2The function f′(x) is given byf′(x)=⎧⎨⎩3x2+2x−10-1≤x<0−sinx0≤x<π/2cosx\pi/2≤x≤π 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 −1≤x≤0 and f′(x)<0 for 0≤x<π/2Therefore, f(x) does not have any extremum at x=0Also f′(x)<0 for 0≤x<π/2 and f′(x)<0 for π/2≤x≤πTherefore, f(x) does not have any extremum at x=π/2

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