The function f(x) = 2 | x | + | x + 2 | – | |x + 2| –2 |x| | has a local minimum or a local maximum at x =

Engineering

Mathematics

Maxima and Minima

Question

$\frac{2}{3}$

$\frac{- 2}{3}$

– 2

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x > 0

f (x) = 2x + x + 2 – | 2 – x |

x $\geq$ 2

2x + x + 2 – x + 2 = 2x + 4

0 $\leq$ x < 2

2x + x + 2 – 2 + x = 4x

x $\leq$ – 2

f (x) = – 2x – x – 2 – | – x – 2 + 2x |

= – 3x – 2 – | x – 2 |

= – 3x – 2 + x – 2 = – 2x – 4

– 2 < x < 0

f (x) = – 2x + x + 2 – | x + 2 + 2x |

= – 2x + x + 2 – | 3x + 2 |

– 2 < x < $\frac{- 2}{3}$ $\frac{- 2}{3}$ < x < 0

– 2x + x + 2 + 3x + 2 – 2x + x + 2 – 3x – 2

2x + 4 – 4x