I am new to proofs and am struggling to prove that $x^3 - x^2 +20$ is bijective, when $x\ge\frac{2}{3}$.
I have shown its not injective when $x$ is a real number, also found the minimum $(\frac{2}{3},\frac{536}{27})$. I assume I need to prove 1) it is injective and 2) it is surjective. Can someone help me set this out? I get as far as $x^3-x^2=x^3-x^2$ and cant simplify any further.
I have tried looking online but all examples are either $f(x)=x^2$ or $f(x)= e^x$ etc. I cant find any with a polynomial in.
The derivative is $$ f'(x)=3x^2-2x=x(3x-2) $$ which is positive for $x>2/3$. So the function is strictly monotonic on the interval $[2/3,\infty)$.
Since $\lim_{x\to\infty}f(x)=\infty$ and $$ f(2/3)=\frac{8}{27}-\frac{4}{9}+20=\frac{560}{27} $$ we know that the range of $f$ over the interval $[2/3,\infty)$ is $[560/27,\infty)$.
Now draw your conclusions.