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Question

Let f:RR be defined as f(x)=|x|+|x21|. The total number of points at which f attains either a local maximum or a local minimum is

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Solution

f(x)=|x|+|x1||x+1|

Case (i) If x1, then
f(x)=x+(x1)(x+1)f(x)=x2+x1

Case (ii) If 0x<1, then
f(x)=x+(1x)(x+1)f(x)=1x2+x

Case (iii) If 1<x<0, then
f(x)=x+(1x)(x+1)f(x)=1x2x

Case (iv) If x1, then
f(x)=x(1x)(x+1)f(x)=x2x1

Plotting the graph for this function


we can see 3 points of local minima and 2 points of local maxima

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