Given such that and for all , is for all ?
(Hint: Use the MVT.)

Question

Given  such that  and  for all  , is  for all  ?
(Hint: Use the MVT.)

Quizplus · Accepted Answer

This question is asking if a certain property holds for all values of a variable. One method for proving this type of statement is to use the Mean Value Theorem (MVT). Since both 11eaa655_3b3a_2a91_a696_47c138e92ea4_TB5596_11 and 11eaa655_3b3a_51a2_a696_af933bc43214_TB5596_11 are given to hold for all 11eaa655_3b3a_51a3_a696_79feb71e4af0_TB5596_11, we can apply the MVT to the function f(x) = 11eaa655_3b3a_51a4_a696_935a637a11df_TB5596_11 on the interval [11eaa655_3b3a_51a5_a696_bb861634353c_TB5596_11, 11eaa655_3b3a_51a3_a696_79feb71e4af0_TB5596_11]. This tells us that there exists a value c in this interval such that f'(c) = (f(11eaa655_3b3a_51a3_a696_79feb71e4af0_TB5596_11) - f(11eaa655_3b3a_51a5_a696_bb861634353c_TB5596_11))/(11eaa655_3b3a_51a3_a696_79feb71e4af0_TB5596_11 - 11eaa655_3b3a_51a5_a696_bb861634353c_TB5596_11). 
By the given property, we know that f'(c) = 11eaa655_3b3a_2a90_a696_65728268ad32_TB5596_11 for all c in the interval [11eaa655_3b3a_51a5_a696_bb861634353c_TB5596_11, 11eaa655_3b3a_51a3_a696_79feb71e4af0_TB5596_11]. Therefore, 11eaa655_3b3a_51a4_a696_935a637a11df_TB5596_11 must also hold for all values of the variable.

Given Such That and for All