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Popular >

extreme f(x)=(x^2-5)^4(x^2+1)^5

Solution

extreme points

f (x) = (x^{2} - 5)^{4} (x^{2} + 1)^{5}

Solution

Minimum (- 5, 0), Maximum (- \frac{7}{3}, \frac{409600000}{19683}), Minimum (0, 625), Maximum (\frac{7}{3}, \frac{409600000}{19683}), Minimum (5, 0)

Solution steps

f^{'} (x) = 6 x (x^{2} + 1)^{4} (x + 5)^{3} (x - 5)^{3} (3 x + 7) (3 x - 7)

Find intervals:

Decreasing

: - \infty < x < - 5,

Increasing

: - 5 < x < - \frac{7}{3},

Decreasing

: - \frac{7}{3} < x < 0,

Increasing

: 0 < x < \frac{7}{3},

Decreasing

: \frac{7}{3} < x < 5,

Increasing

: 5 < x < \infty

Plug

x = - 5

into

(x^{2} - 5)^{4} (x^{2} + 1)^{5} : 0

Minimum (- 5, 0)

Plug

x = - \frac{7}{3}

into

(x^{2} - 5)^{4} (x^{2} + 1)^{5} : \frac{409600000}{19683}

Maximum (- \frac{7}{3}, \frac{409600000}{19683})

Plug

x = 0

into

(x^{2} - 5)^{4} (x^{2} + 1)^{5} : 625

Minimum (0, 625)

Plug

x = \frac{7}{3}

into

(x^{2} - 5)^{4} (x^{2} + 1)^{5} : \frac{409600000}{19683}

Maximum (\frac{7}{3}, \frac{409600000}{19683})

Plug

x = 5

into

(x^{2} - 5)^{4} (x^{2} + 1)^{5} : 0

Minimum (5, 0)

Minimum (- 5, 0), Maximum (- \frac{7}{3}, \frac{409600000}{19683}), Minimum (0, 625), Maximum (\frac{7}{3}, \frac{409600000}{19683}), Minimum (5, 0)

Graph

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Frequently Asked Questions (FAQ)

What is the extreme f(x)=(x^2-5)^4(x^2+1)^5 ?
The extreme f(x)=(x^2-5)^4(x^2+1)^5 is