Question

Consider the function f(x)=6x-cos(x)+5 on the interval 0 is less than or equal to x, and x is less than or equal to 1. The Intermediate Value Theorem guarantees that there is a value c such that f(c)=k for which values of c and k? Fill in the following mathematical statements, giving an interval in non-zero length in each case.

For every k in _______less than or equal to k is less than or equal to _________ there is a c in _________ is less than or equal to c which is less than or equal to________ such as f(c)=k.

Thank you so much! Hopefully someone can figure this out! sorry if the wording is confusing, there are no "less than or equal to" signs on my computer.

Answer 1

Yoiu need f(x) at the endpoints:
f(0) = 4
f(1) = 11-cos(1) = 0.54

For every k in 0.54 <= k <= 4 there is a c in 0 <= c <= 1 such as f(c)=k.

That is, since f(x) changes continuously from 4 to .54, it assume every intermediate value at some point along the interval.