How do I prove this?
Prove that curve
y=x^3+4x
Has one point of inflexion, but no tuning points.
A necessary condition for $\displaystyle x$ to be an inflection point is that $\displaystyle f''(x)=0$
Since there's only one solution to this equation for this function, there is at most one inflection point.
To prove that there's no turning point, study $\displaystyle f'(x)$ and notice that it does not change sign anywhere.