IIT-JEE 2012 · Paper 1 · Mathematics Q56

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MathematicsInteger+4Question 56 of 60

Let $f:\mathbb R\to\mathbb R$ be defined as $f(x)=|x|+|x^2-1|$ . The total number of points at which $f$ attains either a local maximum or a local minimum is

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Answer and worked solution

Official answer: 5

$f$ is even, so study $x\ge0$ .

$0\le x\le1$ : $f=x+1-x^2$ , with $f'=1-2x$ : increasing to a maximum at $x=\frac12$ ( $f=\frac54$ ), then decreasing.
$x\ge1$ : $f=x+x^2-1$ , increasing.

Extrema. Local minima at $x=0$ ( $f=1$ ; $f$ rises on both sides) and $x=1$ ( $f=1$ ); local maximum at $x=\frac12$ . By symmetry also a minimum at $-1$ and a maximum at $-\frac12$ .

Total: $0,\pm\frac12,\pm1$ , five points.

Answer: 5

Topics:Maxima and minima Inequalities and modulus

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IIT-JEE 2012 · Paper 1, Mathematics question 56