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Popular >

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

Solution

extreme points

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

Solution

Minimum (- 15, - 9703125), Saddle (0, 0), Maximum (\frac{5 ( 7 - 19 )}{3}, \frac{1066375000 19 - 4609000000}{729}), Minimum (\frac{5 ( 7 + 19 )}{3}, \frac{- 4609000000 - 1066375000 19}{729})

Solution steps

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

Find intervals:

Decreasing

: - \infty < x < - 15,

Increasing

: - 15 < x < 0,

Increasing

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

Decreasing

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

Increasing

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

Plug

x = - 15

into

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

Minimum (- 15, - 9703125)

Plug

x = 0

into

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

Saddle (0, 0)

Plug

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

into

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

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

Plug

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

into

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

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

Minimum (- 15, - 9703125), Saddle (0, 0), Maximum (\frac{5 ( 7 - 19 )}{3}, \frac{1066375000 19 - 4609000000}{729}), Minimum (\frac{5 ( 7 + 19 )}{3}, \frac{- 4609000000 - 1066375000 19}{729})

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Frequently Asked Questions (FAQ)

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