Find the range of the values for $\displaystyle c$ for which $\displaystyle f(x) = x^3 - 3x^2 + 3cx + 1$ is strictly increasing.
Use the quadratic formula, pay attention in particular to the discriminant.
Think about what it means to have f'(x) > 0. It means if you graph f'(x), you should have every single point above the x axis => no real roots.
So under what conditions does a quadratic equation not have real roots?
but tell me one thing....here we are considering the non-existence of any real root of the quadratic equation because we have a $\displaystyle > 0$ condition. but in case we had the function decreasing, then also we would have considered the same criterion, that is, the non-existence of any real root; and hence the final answer (ie, c > 1) would be the same.
how can you explain this?