Relation between inequalities for first and second derivatives

[nikozm]

Hi,

If f'(x) >= f'(y) can we say that f''(x) >= f''(y) also holds ? And if yes under which conditions ?

Thanks

 

[Orodruin]

No, no such inference can be generally drawn.

 

[Topolfractal]

A simple counter example is
F(x)= x ^2
G(x)= 1,000,000,000,000,000,000 x
DF = 2x
DG= 1,000,000,000,000,000,000
D^2F = 2
D^2G= 0
For x > 500,000,000,000,000,000

Hi,

If f'(x) >= f'(y) can we say that f''(x) >= f''(y) also holds ? And if yes under which conditions ?

Thanks

is not correct. For anything below those bounds it is right. The inequality you stated is only true under certain bounds because the two derivatives vary and may periodically become greater or smaller as in the sine and cosine functions. Bounds the inequality is true under need to be specified , to answer questions about the examples of the inequality you stated. Also because the derivative is a function it varies inequalities can't though and become false for certain values of the function. (The sine cosine remark serves as a good example, but I'm leaning towards arbitrary functions varying in all different manners not just varying periodically.)

 

[mathwonk]

do you know about taylor series? In that series representation of a function, the first derivative is the linear coefficient and the second derive active is (twice) the quadratic coefficient. Your question is thus sort of like asking whether two taylor writes which have the same linear coefficient must also have the same quadratic coefficient. Of course not, as the second coefficient can be anything.

Or geometrically, the first derivative measures the slope of the graph while the second derivative measures the convexity. so this is like asking whether two graphs with the same slope must also have the same convexity, or concavity at a point. can you draw a counterexample?