Suppose that and that for all of t. Must for all of t??

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- December 10th 2009, 07:41 PMIfe[SOLVED] Derivatives of functions
Suppose that and that for all of t. Must for all of t??

- December 10th 2009, 07:45 PMVonNemo19
- December 10th 2009, 08:18 PMIfe
- December 11th 2009, 03:05 AMmr fantastic
- December 13th 2009, 08:10 AMIfe
- December 13th 2009, 08:11 AMIfe
- December 13th 2009, 08:38 AMHallsofIvy
Yes. By the mean value theorem, two functions that have the same derivative must differ by a constant:

Let h(x)= f(x)- g(x). f'(t)= g'(t), then h'(x)= 0 for all t. By the mean value theorem, h(b)- h(a)/(b- a)= 0 so h(b)- h(a)= 0 and h(b)- h(a)= 0. That is, h is a constant and so f and g differ by a constant.

Since f(t)= 5t has f'(x)= 5 for all t,any function with derivative always equal to 5 must be of the form g(x)= 5x+ C. In order that g(0)= -7, C must be -7.