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Popular >

extreme f(x)=(x-2)(x-3)^2(x-4)^3

Solution

extreme points

f (x) = (x - 2) (x - 3)^{2} (x - 4)^{3}

Solution

Minimum (\frac{17 - 13}{6}, - 12813 + \frac{186481 13 - 1680203}{1458} + 1152), Maximum (3, 0), Minimum (\frac{17 + 13}{6}, 12813 + \frac{- 1680203 - 186481 13}{1458} + 1152), Saddle (4, 0)

Solution steps

f^{'} (x) = 2 (x - 4)^{2} (x - 3) (3 x^{2} - 17 x + 23)

Find intervals:

Decreasing

: - \infty < x < \frac{17 - 13}{6},

Increasing

: \frac{17 - 13}{6} < x < 3,

Decreasing

: 3 < x < \frac{17 + 13}{6},

Increasing

: \frac{17 + 13}{6} < x < 4,

Increasing

: 4 < x < \infty

Plug

x = \frac{17 - 13}{6}

into

(x - 2) (x - 3)^{2} (x - 4)^{3} : - 12813 + \frac{186481 13 - 1680203}{1458} + 1152

Minimum (\frac{17 - 13}{6}, - 12813 + \frac{186481 13 - 1680203}{1458} + 1152)

Plug

x = 3

into

(x - 2) (x - 3)^{2} (x - 4)^{3} : 0

Maximum (3, 0)

Plug

x = \frac{17 + 13}{6}

into

(x - 2) (x - 3)^{2} (x - 4)^{3} : 12813 + \frac{- 1680203 - 186481 13}{1458} + 1152

Minimum (\frac{17 + 13}{6}, 12813 + \frac{- 1680203 - 186481 13}{1458} + 1152)

Plug

x = 4

into

(x - 2) (x - 3)^{2} (x - 4)^{3} : 0

Saddle (4, 0)

Minimum (\frac{17 - 13}{6}, - 12813 + \frac{186481 13 - 1680203}{1458} + 1152), Maximum (3, 0), Minimum (\frac{17 + 13}{6}, 12813 + \frac{- 1680203 - 186481 13}{1458} + 1152), Saddle (4, 0)

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Frequently Asked Questions (FAQ)

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