# Nature of stationary point

• Mar 20th 2010, 08:11 PM
yeoky
Nature of stationary point
Find the coordinates of the stationary points on the curve y=x(x-2)^4. Determine the nature of the stationary points.

I am able to find the stationary points are at x=2 and x=0.4

When x=0.4, the second derivative of y=-20.48, hence it is a max point.

When x=2, the second derivative of y =0, which means it is point of inflexion. However, when I plotted the graph of y, I realise that it is a minimum point. Which is correct?
• Mar 20th 2010, 08:18 PM
Prove It
Quote:

Originally Posted by yeoky
Find the coordinates of the stationary points on the curve y=x(x-2)^4. Determine the nature of the stationary points.

I am able to find the stationary points are at x=2 and x=0.4

When x=0.4, the second derivative of y=-20.48, hence it is a max point.

When x=2, the second derivative of y =0, which means it is point of inflexion. However, when I plotted the graph of y, I realise that it is a minimum point. Which is correct?

The second derivative test is INCONCLUSIVE if the second derivative = 0 at that stationary point. You need to check the gradients at points close to the stationary point and show that it goes from positive to negative or vice versa.

Also,
• Mar 26th 2010, 08:06 PM
yeoky
Quote:

Originally Posted by Prove It
The second derivative test is INCONCLUSIVE if the second derivative = 0 at that stationary point. You need to check the gradients at points close to the stationary point and show that it goes from positive to negative or vice versa.

Also,

Thanks (Nod)
• Mar 27th 2010, 03:35 AM
HallsofIvy
The fact that the second derivative is 0 does NOT mean it is an inflection point. An inflection point is where the second derivative changes sign. Of course, for a smooth function, that can only happen where the second derivative is 0 but the converse is not necessarily true.