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受欢迎的函数 >

extreme f(x)=(x+1)^2(x-4)^5(x-7)^4

解答

极值点

f (x) = (x + 1)^{2} (x - 4)^{5} (x - 7)^{4}

解答

极大值 (- 1, 0), 极小值 (\frac{32 - 969}{11}, \frac{- 5.99965 E 14 969 - 1.74164 E 16}{285311670611} - \frac{1024 ( 538920 969 - 18340105 )}{14641} - 2458624), 鞍点 (4, 0), 极大值 (\frac{32 + 969}{11}, \frac{5.99965 E 14 969 - 1.74164 E 16}{285311670611} - \frac{1024 ( - 18340105 - 538920 969 )}{14641} - 2458624), 极小值 (7, 0)

求解步骤

f^{'} (x) = (x - 4)^{4} (x - 7)^{3} (x + 1) (11 x^{2} - 64 x + 5)

Find intervals:

递增

: - \infty < x < - 1,

递减

: - 1 < x < \frac{32 - 969}{11},

递增

: \frac{32 - 969}{11} < x < 4,

递增

: 4 < x < \frac{32 + 969}{11},

递减

: \frac{32 + 969}{11} < x < 7,

递增

: 7 < x < \infty

将

x = - 1

代入

(x + 1)^{2} (x - 4)^{5} (x - 7)^{4} : 0

极大值 (- 1, 0)

将

x = \frac{32 - 969}{11}

代入

(x + 1)^{2} (x - 4)^{5} (x - 7)^{4} : \frac{- 5.99965 E 14 969 - 1.74164 E 16}{285311670611} - \frac{1024 ( 538920 969 - 18340105 )}{14641} - 2458624

极小值 (\frac{32 - 969}{11}, \frac{- 5.99965 E 14 969 - 1.74164 E 16}{285311670611} - \frac{1024 ( 538920 969 - 18340105 )}{14641} - 2458624)

将

x = 4

代入

(x + 1)^{2} (x - 4)^{5} (x - 7)^{4} : 0

鞍点 (4, 0)

将

x = \frac{32 + 969}{11}

代入

(x + 1)^{2} (x - 4)^{5} (x - 7)^{4} : \frac{5.99965 E 14 969 - 1.74164 E 16}{285311670611} - \frac{1024 ( - 18340105 - 538920 969 )}{14641} - 2458624

极大值 (\frac{32 + 969}{11}, \frac{5.99965 E 14 969 - 1.74164 E 16}{285311670611} - \frac{1024 ( - 18340105 - 538920 969 )}{14641} - 2458624)

将

x = 7

代入

(x + 1)^{2} (x - 4)^{5} (x - 7)^{4} : 0

极小值 (7, 0)

极大值 (- 1, 0), 极小值 (\frac{32 - 969}{11}, \frac{- 5.99965 E 14 969 - 1.74164 E 16}{285311670611} - \frac{1024 ( 538920 969 - 18340105 )}{14641} - 2458624), 鞍点 (4, 0), 极大值 (\frac{32 + 969}{11}, \frac{5.99965 E 14 969 - 1.74164 E 16}{285311670611} - \frac{1024 ( - 18340105 - 538920 969 )}{14641} - 2458624), 极小值 (7, 0)

作图

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