Differentiation Question (1 Viewer)

[Crosswinds]

Okie, this question's annoying me 'cause i know it's really easy but I can't get the same answer as is in the back of the textbook:

"Find any turning points on the curve y = (4x^2 - 1)^4 and determine their nature."

I worked out that the stationary points are (0, 1), (0.5, 0) and (-0.5, 0), and that the first one of those [(0, 1)] is a maximum turning point. All of which is correct according to the answers. But then I found that the other two were points of inflexion, whereas the answers say (0.5, 0) and (-0.5, 0) are both minimum TPs.

Could someone please explain what I'm doing wrong?? Thanks!

 

[Trebla]

Okie, this question's annoying me 'cause i know it's really easy but I can't get the same answer as is in the back of the textbook:

"Find any turning points on the curve y = (4x^2 - 1)^4 and determine their nature."

I worked out that the stationary points are (0, 1), (0.5, 0) and (-0.5, 0), and that the first one of those [(0, 1)] is a maximum turning point. All of which is correct according to the answers. But then I found that the other two were points of inflexion, whereas the answers say (0.5, 0) and (-0.5, 0) are both minimum TPs.

Could someone please explain what I'm doing wrong?? Thanks!

Remember that you have to check that there is a CHANGE IN SIGN in the second derivative to have an inflexion point.
dy/dx = 32x(4x² - 1)³
d²y/dx² = 32(4x² - 1)³ + 32x(24x)(4x² - 1)²
= 32(4x² - 1)²[4x² - 1 + 24x²]
= 32(4x² - 1)²(28x² - 1)
Since x = ±0.5 makes both dy/dx and d²y/dx² as zero, we can only deduce they horizontal points of inflexion if we test that the signs change on either side of ±0.5. When you try this, it turns out that d²y/dx² DOES NOT change sign, therefore our results are inconclusive. This can happen when you have polynomials of degree 4 or higher.

What you should do is test both sides x = ±0.5 and check the sign of the first derivative (that table thingy). e.g. test x = 0, and x = ± 1. A table of values of dy/dx at those x-values should tell you it's a minimum.

 

[Crosswinds]

O yay it works! Thankyou!