Higher derivatives

Question

Prove that the curve y=3x^4+kx has only one turning point.

Answer 1

y' = 12x³ + k

 

y' = 0 only if x = cube root of (-k/12)  so there is only one critical point.  To its left, the y' values are all negative, IF k>0; to its right, y' vaules are all positive IF k<0.  

 

(This behavior is reversed if k < 0.) So there is either decreasing/increasing behavior (or its reverse). Therefore only one turning point.

 

y" = 36x² so y" = 0 only at x=0.  To the left or right of 0, the concavity is always upward since y" > 0, so there's no inflection point.