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 which f>g for all x>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.
Originally posted by AcolyteThis is clever, but isn't it only of interest to 'mathematical types' or maybe Thomas Malthus 😉?
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.
Originally posted by AcolyteJust 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.
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.
Originally posted by Acolytemust...learn....cal..cu...lus....
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.