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Find the local extremum values of the functions: f(x) = – (x – 1)3(x + 1)2 |
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Answer» Given as f(x) = – (x – 1)3(x + 1)2 f’(x) = – 3(x – 1)2(x + 1)2 – 2(x – 1)3(x + 1) = – (x – 1)2(x + 1) (3x + 3 + 2x – 2) = – (x – 1)2(x + 1) (5x + 1) f’’(x) = – 2(x – 1)(x + 1)(5x + 1) – (x – 1)2(5x + 1) – 5(x – 1)2(x – 1) For the maxima and minima, f'(x) = 0 – (x – 1)2(x + 1) (5x + 1) = 0 x = 1, – 1, – 1/5 f’’ (1) = 0 x = 1 is the inflection point f’’(– 1) = – 4× – 4 = 16 > 0 x = – 1 is the point of minima f’’ (– 1/5) = – 5(36/25) × 4/5 = – 144/25 < 0 x = – 1/5 is the point of maxima Thus, local max value = f (– 1/5) = 3456/3125 The local min value = f (– 1) = 0 |
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