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Popular >

extreme (x-7)(x-5)^2(x-3)(x-1)

Solution

extreme points

(x - 7) (x - 5)^{2} (x - 3) (x - 1)

Solution

Maximum (1.65442 \dots, 52.68703 \dots), Minimum (3.68776 \dots, - 10.54328 \dots), Maximum (5, 0), Minimum (6.45780 \dots, - 21.74567 \dots)

Solution steps

Find the critical points:

x \approx 1.65442 \dots, x \approx 3.68776 \dots, x = 5, x = 6.45780 \dots

Domain of

(x - 7) (x - 5)^{2} (x - 3) (x - 1) : - \infty < x < \infty

Find intervals:

Increasing

: - \infty < x < 1.65442 \dots,

Decreasing

: 1.65442 \dots < x < 3.68776 \dots,

Increasing

: 3.68776 \dots < x < 5,

Decreasing

: 5 < x < 6.45780 \dots,

Increasing

: 6.45780 \dots < x < \infty

Plug the extreme point

x = 1.65442 \dots

into

(x - 7) (x - 5)^{2} (x - 3) (x - 1) \Rightarrow y = 52.68703 \dots

Maximum (1.65442 \dots, 52.68703 \dots)

Plug the extreme point

x = 3.68776 \dots

into

(x - 7) (x - 5)^{2} (x - 3) (x - 1) \Rightarrow y = - 10.54328 \dots

Minimum (3.68776 \dots, - 10.54328 \dots)

Plug the extreme point

x = 5

into

(x - 7) (x - 5)^{2} (x - 3) (x - 1) \Rightarrow y = 0

Maximum (5, 0)

Plug the extreme point

x = 6.45780 \dots

into

(x - 7) (x - 5)^{2} (x - 3) (x - 1) \Rightarrow y = - 21.74567 \dots

Minimum (6.45780 \dots, - 21.74567 \dots)

Maximum (1.65442 \dots, 52.68703 \dots), Minimum (3.68776 \dots, - 10.54328 \dots), Maximum (5, 0), Minimum (6.45780 \dots, - 21.74567 \dots)

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