What is the extreme f(x)=(3x+1)^5(2x+3)^2(2-x)^3 ?

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

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

extreme points

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

Maximum (- \frac{3}{2}, 0), Minimum (\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), Saddle (- \frac{1}{3}, 0), Maximum (\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), Saddle (2, 0)

Solution steps

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

Find intervals:

Increasing

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

Decreasing

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

Increasing

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

Increasing

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

Decreasing

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

Decreasing

: 2 < x < \infty

Plug

x = - \frac{3}{2}

into

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

Maximum (- \frac{3}{2}, 0)

Plug

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

into

(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

Minimum (\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)

Plug

x = - \frac{1}{3}

into

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

Saddle (- \frac{1}{3}, 0)

Plug

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

into

(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

Maximum (\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)

Plug

x = 2

into

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

Saddle (2, 0)

Maximum (- \frac{3}{2}, 0), Minimum (\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), Saddle (- \frac{1}{3}, 0), Maximum (\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), Saddle (2, 0)

y = 2 x^{3} - 12 x^{2} - 270 x + 2

y = 0.0004 x^{2} - 0.0118 x + 2.6821

x (\frac{8}{5} x^{3} - 2 x^{2})

f (x) = ln (\frac{4 - x}{2 + 2 x})

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

What is the extreme f(x)=(3x+1)^5(2x+3)^2(2-x)^3 ?
The extreme f(x)=(3x+1)^5(2x+3)^2(2-x)^3 is