Question

Identify a counterexample to disprove n3 ≤ 3n2, where n is a real number.

Answer 1

$n^3\le3n^2\iff n^3-3n^2\le0\iff n^2(n-3)\le0$

The quantity on the left hand side will be positive for any $n>3$, so suppose consider $n=4$.

Now,

$4^3=64$

but

$3(4)^2=48$

and $64\not\le48$

Answer 2

$n^3-3n^2\ \leq \ 0$    $\text{Move all terms to one side}$ 

$n^2(n-3) \leq0$   $\text{Factor out the common term} \ {n}^{2}$ 

$n(n-3) = 0$  $\text{when n=0,3}$ 

⇒ $\text{From the values of} \ n^2 \text{ above, we have these 3 intervals to test}$ 

⇒ $n \leq0$ 

⇒ $0 \leq n \leq 3$ 

⇒ $n \geq 3$

⇒ $\text{Pick a test point for each interval}$