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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).

I'm new to calculus. Appreciate your answers.

Thanks,

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First question : You can tell whether it's concave up (convexe) or concave down (concave) using the variation of the first derivative which is easily checked using the second derivative. But if you know how varies the first derivative, that is increasing or decreasing, then it is sufficient to check concavity.

Second question : I'm not sure to understand your question, I think you are confused. The sign of the second derivative of a function over a certain interval will show that the initial function is either concave or convexe over the same interval.

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.
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