Derivatives

Acolyte
Posers and Puzzles 24 Mar '04 00:37
1. Acolyte
24 Mar '04 00:375 edits
Suppose f and g are two infinitely differentiable functions on the reals, and suppose that, for some real A, k&gt;0, n&gt;0, and for all x&gt;A:
f^(n)(x) - g^(n)(x) &gt; k (where ^(n) means nth derivative.)

Show that there exists a number B so that for all x&gt;B, f(x) - g(x) &gt; 0. Deduce that, if f is e^qx for q positive, and g is a polynomial, there exists a B for which f&gt;g for all x&gt;B.

Once again I ask mathematical types not to call out the answer until others have had a chance to think about it. This one's off the top of my head, so I apologise if people find it too easy or hard, or if I've got the problem wrong.

Note to the non-mathematical: a derivative is a 'rate of change', and the derivative of an nth derivative is a (n+1)th derivative. So the second derivative of f is 'rate of change of rate of change of f', and so on.
2. royalchicken
CHAOS GHOST!!!
24 Mar '04 18:00
Originally posted by Acolyte
Suppose f and g are two infinitely differentiable functions on the reals, and suppose that, for some real A, k>0, n>0, and for all x>A:
f^(n)(x) - g^(n)(x) > k (where ^(n) means nth derivative.)

Show that there exists a number B so that for all x>B, f(x) - g(x) > 0. Deduce that, if f is e^qx for q positive, and g is a polynomial, there exists a B for w ...[text shortened]... ivative. So the second derivative of f is 'rate of change of rate of change of f', and so on.
This is clever, but isn't it only of interest to 'mathematical types' or maybe Thomas Malthus ðŸ˜‰?

3. Acolyte
25 Mar '04 18:20
Originally posted by Acolyte
Suppose f and g are two infinitely differentiable functions on the reals, and suppose that, for some real A, k>0, n>0, and for all x>A:
f^(n)(x) - g^(n)(x) > k (where ^(n) means nth derivative.)

Show that there exists a number B so that for all x>B, f(x) - g(x) > 0. Deduce that, if f is e^qx for q positive, and g is a polynomial, there exists a B for w ...[text shortened]... ivative. So the second derivative of f is 'rate of change of rate of change of f', and so on.
Just noticed that f and g need only be n times differentiable. There are a few variations on this theme, and I was messing around with them after posting, hence the edits.
4. Acolyte