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受欢迎的函数 >

extreme f(x)=(3x+1)^5(2x+3)^2(2-x)^3

解答

极值点

f (x) = (3 x + 1)^{5} (2 x + 3)^{2} (2 - x)^{3}

解答

极大值 (- \frac{3}{2}, 0), 极小值 (\frac{1 - 7 109}{60}, \frac{136600387 109 - 1318618861}{12150000} + \frac{5.72682 E 17 - 6.01163 E 16 109}{3.888 E 13} + \frac{4 ( 2761 - 637 109 )}{225} + 72), 鞍点 (- \frac{1}{3}, 0), 极大值 (\frac{1 + 7 109}{60}, \frac{- 1318618861 - 136600387 109}{12150000} + \frac{5.72682 E 17 + 6.01163 E 16 109}{3.888 E 13} + \frac{4 ( 2761 + 637 109 )}{225} + 72), 鞍点 (2, 0)

求解步骤

f^{'} (x) = - (1 + 3 x)^{4} (x - 2)^{2} (2 x + 3) (60 x^{2} - 2 x - 89)

Find intervals:

递增

: - \infty < x < - \frac{3}{2},

递减

: - \frac{3}{2} < x < \frac{1 - 7 109}{60},

递增

: \frac{1 - 7 109}{60} < x < - \frac{1}{3},

递增

: - \frac{1}{3} < x < \frac{1 + 7 109}{60},

递减

: \frac{1 + 7 109}{60} < x < 2,

递减

: 2 < x < \infty

将

x = - \frac{3}{2}

代入

(3 x + 1)^{5} (2 x + 3)^{2} (2 - x)^{3} : 0

极大值 (- \frac{3}{2}, 0)

将

x = \frac{1 - 7 109}{60}

代入

(3 x + 1)^{5} (2 x + 3)^{2} (2 - x)^{3} : \frac{136600387 109 - 1318618861}{12150000} + \frac{5.72682 E 17 - 6.01163 E 16 109}{3.888 E 13} + \frac{4 ( 2761 - 637 109 )}{225} + 72

极小值 (\frac{1 - 7 109}{60}, \frac{136600387 109 - 1318618861}{12150000} + \frac{5.72682 E 17 - 6.01163 E 16 109}{3.888 E 13} + \frac{4 ( 2761 - 637 109 )}{225} + 72)

将

x = - \frac{1}{3}

代入

(3 x + 1)^{5} (2 x + 3)^{2} (2 - x)^{3} : 0

鞍点 (- \frac{1}{3}, 0)

将

x = \frac{1 + 7 109}{60}

代入

(3 x + 1)^{5} (2 x + 3)^{2} (2 - x)^{3} : \frac{- 1318618861 - 136600387 109}{12150000} + \frac{5.72682 E 17 + 6.01163 E 16 109}{3.888 E 13} + \frac{4 ( 2761 + 637 109 )}{225} + 72

极大值 (\frac{1 + 7 109}{60}, \frac{- 1318618861 - 136600387 109}{12150000} + \frac{5.72682 E 17 + 6.01163 E 16 109}{3.888 E 13} + \frac{4 ( 2761 + 637 109 )}{225} + 72)

将

x = 2

代入

(3 x + 1)^{5} (2 x + 3)^{2} (2 - x)^{3} : 0

鞍点 (2, 0)

极大值 (- \frac{3}{2}, 0), 极小值 (\frac{1 - 7 109}{60}, \frac{136600387 109 - 1318618861}{12150000} + \frac{5.72682 E 17 - 6.01163 E 16 109}{3.888 E 13} + \frac{4 ( 2761 - 637 109 )}{225} + 72), 鞍点 (- \frac{1}{3}, 0), 极大值 (\frac{1 + 7 109}{60}, \frac{- 1318618861 - 136600387 109}{12150000} + \frac{5.72682 E 17 + 6.01163 E 16 109}{3.888 E 13} + \frac{4 ( 2761 + 637 109 )}{225} + 72), 鞍点 (2, 0)

作图

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