*Solve the inequality `x^5(x-6)<=0` *

(1) There are only two real zeros for the function: x=0, x=6.

(2) This is a polynomial, so it is continuous everywhere -- no breaks in the graph

(3) This is a sixth degree polynomial with positive leading coefficient, so its end behavior is that it grows without bound as x goes to `+-oo`

(4) So we just have to check a test value in the intervals

(`-oo,0),(0,6),(6,oo)` ;e.g test -1,1,and 7.

At -1, the function is `(-1)^5(-1-6)=(-1)(-7)=7>0`

At 1, the function is `(1)^5(1-6)=(1)(-5)=-5<0`

At 7, the function is `(7)^5(7-6)=7^5>0`

So the function is nonpositive only on `[0,6]` .

We have to solve the inequality: x^5(x - 6) <= 0

x^5(x - 6) <= 0

if either x^5 <= 0 and x - 6 >= 0 or if x^5 >= 0 and x - 6 <= 0

x^5 <= 0 and x - 6 >= 0

=> x <= 0 and x >= 6

this is not satisfied for any values of x

x^5 >= 0 and x - 6 <= 0

=> x>=0 and x <= 6

This is satisfied for values of x that lie in [0, 6]

**The solution of the inequality is [0, 6].**

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