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Question

# The total number of points of local maxima and local minima of the function f(x)=⎧⎪⎨⎪⎩(2+x)3, −3<x≤−1x23, −1<x<2 is

A
0
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B
1
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C
2
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D
3
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Solution

## The correct option is C 2 f(x)=⎧⎪⎨⎪⎩(2+x)3, −3<x≤−1x23, −1<x<2f′(x)=⎧⎪ ⎪⎨⎪ ⎪⎩3(2+x)2, −3<x≤−123x−13, −1<x<2 From f′(x)=0 we get x=−2 and at x=0, f′(x) becomes undefined, and f(x) is not differentiable at x=−1 So, here we get three critical points i.e. x=−2,−1,0 But there is no sign change for f′(x) at x=−2. So, at x=−2, f(x) has neither minimum nor maximum Since, f′(x) changes sign from negative to positive as x crosses 0 from left to right, therefore x=0 is a point of local minima. From the above graph clearly, x=−1 is a point of local maxima because f(x→−1−)<f(−1)>f(x→−1+) The total number of local maximum or minimum =2

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