Differentiable function $f(x)=4x^7 −14x^4 +30x−17$ I am trying to prove that the function $f:\Bbb R→\Bbb R$, $f(x)=4x^7 −14x^4 +30x−17$,is injective. To do this I need to prove it is differentiable from first principles. I can then prove its derivative is strictly increasing to show it is injective. Any help on the proof that it's differentiable would be great, especially in what delta to choose.

Since we can  evaluate a derivative, our function is differentiable. 
Also, $$f'(x)=28x^6-56x^3+30=28\left(x^6-2x^3+\frac{15}{14}\right)=28\left((x^3-1)^2+\frac{1}{14}\right)>0.$$
$$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=$$
$$=\lim_{h\rightarrow0}\frac{4((x+h)^7-x^7)-14((x+h)^4-x^4)+30(x+h-x)}{h}=$$
$$=4(7x^6)-14(4x^3)+30=28x^6-56x^3+30.$$
