If 0<x<1, why is x^4 - x^5 < x^2 - X^3?

Please help. Thanks.

If 0<x<1, why is x^4 - x^5 < x^2 - X^3?

X^4(1-X) < X^2(1-X)
X^2(1-X) < (1-X)
X^2 < 1

If X is greater than 0 but less than 1, you know it is a positive fraction. Any positive fraction < 1, squared, will still be less than 1.

Very good solution by Mikus. I just want to point out that normally dividing both sides of an inequality by a variable amount is dangerous territory because of the possibility that the variable amount could be zero or a negative number. It works out well in this problem because of the constraint that 0<x<1. When we divide both sides by x^2, we know it's a positive, nonzero number. Also, when we divide both sides by (1-x), again we know it's a positive, nonzero number.

Again, nice work, Mikus!

Rey