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Let I RvecI R be defined as f(x)=|x|++x^...

Let `I RvecI R` be defined as `f(x)=|x|++x^2-1|dot` The total number of points at which `f` attains either a local maximum or a local minimum is_______

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The correct Answer is:
5
Plan : (i) Local maximum and local minimum are those points at which `f ' (x) = 0 `, when defined for all real numbers.
(ii) Local maximum and local minimum for piecewise functions are also been checked at sharp edges.
Description of Situation `y = |x| = {{:( x",",, if x ge 0),( - x",",, if x lt 0):}`
Also, `y =|x ^(2) - 1 |= {{:(( x^(2) - 1)",",, if x le - 1 or x ge 1 ),( (1- x ^(2)) ",",, if - 1 le x le 1 ):}`
`y = |x| + |x^(2) - 1 | = {{:( - x+ 1 - x ^(2)",",, if x le - 1 ), ( - x + 1 - x ^(2)",",, if - 1 le x le 0 ), ( x + 1 - x ^(2)",",, if 0 le x le 1 ),( x + x ^(2) - 1 ",",, if x ge 1 ):}`
` " " = {{:( - x ^(2) - x + 1",",, if x le - 1 ), ( - x ^(2) - x + 1",",, if 0 le x le 1), ( x ^(2) + x - 1 ",",, if x ge 1 ):}`
which could be graphically shown as

Thus, ` f (x)` attains maximum at ` x = (1)/(2), (-1)/(2) and f(x) ` attains minimum at ` x = - 1 , 0, 1 `
`rArr ` Total number of points `= 5 `
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