Let k(x) = h(x) - g(x), where g and h are infinitely differentiable functions from R to R. a. Suppose h'(x) = g'(x) for all x. Characterize the possible functions k(x). b. Suppose h''(x) = g''(x) for all x. Characterize the possible functions k(x). c. Suppose h''(x) = g''(x) for all x. Suppose further that k(0) = 0 and |k'(0)|< 1. Find the limit as n goes to infinity of k(x)n for |x|<1. d. Define f(n) to be the nth derivative of a function mapping R to R. Suppose hn(x) = gn(x) for all x. Suppose k(x1) = k(x2) = ... = k(xn) = 0 for x1 < x2 < x3 < ... < xn . Show that g = h.
Let k(x) = h(x) - g(x), where g and h are infinitely
a. Suppose h'(x) = g'(x) for all x. Characterize the possible functions k(x).
b. Suppose h''(x) = g''(x) for all x. Characterize the possible functions k(x).
c. Suppose h''(x) = g''(x) for all x. Suppose further that k(0) = 0 and |k'(0)|< 1. Find the limit as n goes to infinity of k(x)n for |x|<1.
d. Define f(n) to be the nth derivative of a function mapping R to R. Suppose hn(x) = gn(x) for all x. Suppose k(x1) = k(x2) = ... = k(xn) = 0 for
x1 < x2 < x3 < ... < xn . Show that g = h.
By bartleby rules, only first three sub-parts have been answered
Given where and are infinitely differentiable functions from to
Suppose,
To characterize possible functions
Since, , therefore,
So, when , in that case the function is characterized as a constant function.
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