Second derivative

When $\frac{d^{2}y}{dx^{2}} = 0$ there may be a point of inflexion which means that the graph seems to 'level off' at these points.

However, when $\frac{d^{2}y}{dx^{2}} = 0$ there isn't strictly going to be a point of inflexion, it could also be a MAXimum point or a MINimum point. This can be checked by selecting x values to the right and the left of the point where $\frac{d^{2}y}{dx^{2}} = 0$ and checking what the graph would look like.

Does that make sense?

but does that mean for the 2nd derivative to be zero the first derivative must also be 0?

Yes! The first derivative is 0 if it is a stationary point. You use the secondary derivative to find out the nature of said stationary point

Yes! The first derivative is 0 if it is a stationary point. You use the secondary derivative to find out the nature of said stationary point

thanks

You can have first derivative non zero and second derivative zero.
This can produces a non stationary point of inflexion.
EVERY cubic has a point of inflexion, not necessarily stationary.

Thanks.

You can have first derivative non zero and second derivative zero.
This can produces a non stationary point of inflexion.
EVERY cubic has a point of inflexion, not necessarily stationary.

isn't a point of inflexion a stationary point?- gradient 0

A point of inflexion maybe stationary or non stationary.

The best way to picture what a point of inflexion is to think of it as the point where the tangent to the curve at that point crosses the curve itself.
For any continuous curve, between a local max and a local min there must be a point of inflexion. (not necessarily stationary)

If stationary (gradient zero) then it has an additional property and that is a stationary point of inflexion

ok but going back to the original question how at x=4 can dy/dx=0 and at 2 and 6 d2y/dx2=0