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受欢迎的函数 >

extreme x^6-10x^5-400x^4+2500x^3

解答

极值点

x^{6} - 10 x^{5} - 400 x^{4} + 2500 x^{3}

解答

极小值 (- 15, - 9703125), 鞍点 (0, 0), 极大值 (\frac{5 ( 7 - 19 )}{3}, \frac{1066375000 19 - 4609000000}{729}), 极小值 (\frac{5 ( 7 + 19 )}{3}, \frac{- 4609000000 - 1066375000 19}{729})

求解步骤

f^{'} (x) = 6 x^{5} - 50 x^{4} - 1600 x^{3} + 7500 x^{2}

Find intervals:

递减

: - \infty < x < - 15,

递增

: - 15 < x < 0,

递增

: 0 < x < \frac{5 ( 7 - 19 )}{3},

递减

: \frac{5 ( 7 - 19 )}{3} < x < \frac{5 ( 7 + 19 )}{3},

递增

: \frac{5 ( 7 + 19 )}{3} < x < \infty

将

x = - 15

代入

x^{6} - 10 x^{5} - 400 x^{4} + 2500 x^{3} : - 9703125

极小值 (- 15, - 9703125)

将

x = 0

代入

x^{6} - 10 x^{5} - 400 x^{4} + 2500 x^{3} : 0

鞍点 (0, 0)

将

x = \frac{5 ( 7 - 19 )}{3}

代入

x^{6} - 10 x^{5} - 400 x^{4} + 2500 x^{3} : \frac{1066375000 19 - 4609000000}{729}

极大值 (\frac{5 ( 7 - 19 )}{3}, \frac{1066375000 19 - 4609000000}{729})

将

x = \frac{5 ( 7 + 19 )}{3}

代入

x^{6} - 10 x^{5} - 400 x^{4} + 2500 x^{3} : \frac{- 4609000000 - 1066375000 19}{729}

极小值 (\frac{5 ( 7 + 19 )}{3}, \frac{- 4609000000 - 1066375000 19}{729})

极小值 (- 15, - 9703125), 鞍点 (0, 0), 极大值 (\frac{5 ( 7 - 19 )}{3}, \frac{1066375000 19 - 4609000000}{729}), 极小值 (\frac{5 ( 7 + 19 )}{3}, \frac{- 4609000000 - 1066375000 19}{729})

作图

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