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Popular >

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

Solution

extreme points

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

Solution

Minimum (\frac{5 - 13}{6}, - \frac{143 13 + 587}{1458}), Maximum (1, 0), Minimum (\frac{5 + 13}{6}, \frac{143 13 - 587}{1458}), Saddle (2, 0)

Solution steps

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

Find intervals:

Decreasing

: - \infty < x < \frac{5 - 13}{6},

Increasing

: \frac{5 - 13}{6} < x < 1,

Decreasing

: 1 < x < \frac{5 + 13}{6},

Increasing

: \frac{5 + 13}{6} < x < 2,

Increasing

: 2 < x < \infty

Plug

x = \frac{5 - 13}{6}

into

x (x - 1)^{2} (x - 2)^{3} : - \frac{143 13 + 587}{1458}

Minimum (\frac{5 - 13}{6}, - \frac{143 13 + 587}{1458})

Plug

x = 1

into

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

Maximum (1, 0)

Plug

x = \frac{5 + 13}{6}

into

x (x - 1)^{2} (x - 2)^{3} : \frac{143 13 - 587}{1458}

Minimum (\frac{5 + 13}{6}, \frac{143 13 - 587}{1458})

Plug

x = 2

into

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

Saddle (2, 0)

Minimum (\frac{5 - 13}{6}, - \frac{143 13 + 587}{1458}), Maximum (1, 0), Minimum (\frac{5 + 13}{6}, \frac{143 13 - 587}{1458}), Saddle (2, 0)

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Frequently Asked Questions (FAQ)

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