I have proved this by induction but with direct proof, I'm not really sure where to start. Thank you.
Give a direct proof that, $n^3 > n^2 − 6n + 4$ for all $n ∈ {\mathbb N}$ with $n ≥ 2$ .
I have proved this by induction but with direct proof, I'm not really sure where to start. Thank you.
Give a direct proof that, $n^3 > n^2 − 6n + 4$ for all $n ∈ {\mathbb N}$ with $n ≥ 2$ .
For what is worth: $n^3-n^2+6n-4>0$
$n^3-3n^2+3n-1+2n^2+3n-3>0$
$(n-1)^3+2n^2+3(n-1)>0$
$n^3-n^2+6n-4=(n^3-n^2+6n-6)+2=(n-1)(n^2+6)+2>0$ for $n>1$.
For $n \geq 1$
$n^3 \geq n^2$ $6n > 4$
add them together.
Since $n\geq 2$ we can say that $n^3 \geq 2n^2=n^2+n^2\geq n^2+4 >n^2-6n+4$. So $n^3>n^2-6n+4$
Let $f(x)=x^3-(x^2-6x+4)=x^3-x^2+6x-4$. This is a polynomial, therefore a differentiable function on $\mathbb{R}$. Its derivative is given by $f'(x)=3x^2-2x+6.$ The discriminant of the quadratic polynomial is $4-4\cdot 18<0$, thus its roots are imaginary. In particular, since $f'(0)=6$, the derivative $f'(x)>0$ for all $x\in\mathbb{R}$. This implies that $f(x)$ is a strictly increasing function for all $x\in\mathbb{R}$. Since $f(1)=2$, we must have that $f(x)\geq 2>0$ for all $x\geq 1$. In particular, $f(x)>0$ for all $x\geq 1$, and $x^3>x^2-6x+4$ for all $x \geq 1$.