# Fundamental Calculus Doubts - Differentiation

With the first derivative, you can tell where, on the original function, the gradient is increasing or decreasing. Further, you use the second derivative for gauging whether a given interval is concave up or concave down.

My question:

*-*Is it possible to tell whether a graph is concave up/ concave down just from the first derivative? If that's the case; why work out the second derivative?

*-*When you work out where on a function (through the second derivative) the intervals are concave up/ concave down; is it the same intervals that are concave up/ concave down on the original function?

*-*Also, with the first derivative, you can work out the stationary points on the original function, right. Would both the derivative and original function have the same points of 'x' (the stationary points, that is).

Thanks,

Third question : Say you want to study a function $f$ and want to check the stationary points, that is maximum, minimum and saddle points. These points have all in common a flat tangent line, that is tangent of equation $y = k$ for some constant $k$.
When the tangent is flat, this means that the first derivative is zero at this point. So if $f'(a) = 0$, this means that $f(a)$ is either a maximum, a minimum or a saddle point. Then calculate the second derivative at $x=a$, if :
• $f''(a) > 0$, then you have a minimum.
• $f''(a) < 0$, then you have a maximum.
• $f''(a) = 0$, then you can't tell.