Why does the second derivative fail for y=x^4 and other similar functions? (1 Viewer)

[Andy03]

Hey guys,

I was wondering why the second derivative gives me an incorrect answer to this question:
'Find the coordinates and nature of the stationary points on the curve y= x4.'

Knowing the graph for the curve, the answer is obviously (0,0) as a minimum turning point.

I've obtained the stationary point (0,0) from the first derivative, and now have the choice to test its nature with either the first or second derivative. Whereas the first gives me the minimum turning point, the second gives me a horizontal point of inflection, as subbing x= 0 into y''gives 0. Why does this happen?

Thanks for any help!

 

[awesome-0_4000]

The second derivative doesn't fail. A point of inflection occurs when there is a change of concavity. To test this, you substitute a point on either side of the point that makes the second derivative zero. Substituting the points -1 and 1 into the second derivative of y=x^4 gives no sign change, meaning that it is not a point of inflection.

 

[Andy03]

Oh ok. Would I have to test surrounding points for every stationary point tested this way?

 

[rumbleroar]

You can use the table method so you check values on either side of your "x" when f'(x)=0