(x-2)(x-3)^4/(x-1)^5≤0

Предыдущий
вопрос Следующий
вопрос

To solve this inequality, we need to first find the critical points by setting the numerator and denominator equal to zero:

Critical points from the numerator:
(x - 2) = 0 => x = 2
(x - 3) = 0 => x = 3

Critical points from the denominator:
(x - 1) = 0 => x = 1

Now, let's plot these critical points on a number line and test the intervals between them:

Test (0, 1):
Choose x = 0:
(-2)(-3)^4/(-1)^5 = -162/(-1) = 162 > 0

Test (1, 2):
Choose x = 1.5:
(-0.5)(-1.5)^4/(-0.5)^5 = 0.5(5.06)/(0.03125)
= 2.53/0.03125 > 0

Test (2, 3):
Choose x = 2.5:
(0.5)(-0.5)^4/(-1.5)^5 = 0.5*0.0625/(-0.0791) = 0.03125/(-0.0791) < 0

Test (3, ∞):
Choose x = 4:
(2)(1)^4/2^5 = 2/32 = 1/16 > 0

Therefore, the solution to the inequality (x-2)(x-3)^4/(x-1)^5 ≤ 0 is:
x ∈ [2, 3]