Sketching gradient functions of a graph. (1 Viewer)

[Forbidden.]

I haven't properly learned the technique to draw a gradient function of f(x).
But I assume f'(x) is the gradient function of f(x), so f"(x) must be a gradient function of f'(x) if I'm correct.


Let's assume there is point A, B & C on f(x) as f(A), f(B), f(C), f(D) & f(E).

If f(A) is a minimum point then f'(A) is zero
If f(B) is a maximum point then f'(B) is zero
If f(C) is a rising point of inflexion then f'(C) must be a maximum point ?
if f(D) is a falling point of inflexion then f'(D) must be a minimum point ?

Am I correct with all of these ?

Then say f(E) is a horizontal point of inflexion, what is the nature/type of point at f'(E) ?

 

[airie]

I haven't properly learned the technique to draw a gradient function of f(x).
But I assume f'(x) is the gradient function of f(x), so f"(x) must be a gradient function of f'(x) if I'm correct.


Let's assume there is point A, B & C on f(x) as f(A), f(B), f(C), f(D) & f(E).

If f(A) is a minimum point then f'(A) is zero
If f(B) is a maximum point then f'(B) is zero

All above are correct.

If f(C) is a rising point of inflexion then f'(C) must be a maximum point ?
if f(D) is a falling point of inflexion then f'(D) must be a minimum point ?

These are wrong. If the point at C is a rising point of inflexion, then the gradient of the graph first decreases till at C, then increases. Thus the point will be a minimum on the graph of f'(x). Similarly, the point will be a maximum on f'(x), when it is a decreasing point of inflexion on f(x).

Then say f(E) is a horizontal point of inflexion, what is the nature/type of point at f'(E) ?

Depends on how the gradient changes. If the gradient first decreases to the point, then increases (think of x=0 on the graph of x³), the point will be a minimum on f'(x). If the gradient first increases to zero at the point and then decreases ((think of x=0 on the graph of -x³), the point will be a maximum on f'(x).

 

[Forbidden.]

So I just have to remember -/+x² and -/+ x³ ?