Last question for tonight. i need solving this problem.

consider the function f(x)=5x−cos(x)+2 on the interval 0≤x≤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 with non-zero length in each case.

i've included a picture for reference.

Intermediate value theorem
(j. stewart, “calculus”, 7th ed., brook/coles.)

if you have a function  that's is continuous on a closed interval  , and a number  is an intermediate value between  and  , then there's some number  in the interval  that satisfies .



the c is for  the x-value or the input value into the function.
the k is for the y-value or the output value of the function at c.

so in this question, we have a continuous function  on  . the interval in question for the ivt must be  then.

k is between the values of the function at the endpoints. so 

so for  , we need to get the values of the function at the endpoints of the interval to determine the range that k can be in.

   

   

and c must be a number in the interval  . (your current answers of  goes out of the interval originally stated for the function. the c is the x-value or the input value into the function.)

so for every    in  , there is a  in  etc.. though i don't know about how many decimal places to round to  for