intermediate value theorem with trigonometry

[Dorian Gray]

Greetings Mathematicians,

I am having some issues with this question.

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval

cos x = x (0,1)


Am I suppose to "plug" 0 and 1 in for x? Am I supposed to rearrange the equation to cos x - x = 0 ? I am not sure how to address this problem

Thank you.

 

[Dorian Gray]

Thank you JeffM for your response. Here is a copy of my work



Does it look like I am on the right track?

 

[Dorian Gray]

Thank you

Thanks for the further explanation. Besides this problem, would it then be best to look at a graph?

 

[Dorian Gray]

thank you

Thank you very much JeffM. You provided plenty of information that went above and beyond. My professor (Calculus I) said that the only type of intermediate value theorem problem on our upcoming exam will be the case where one answer will be positive and one will be negative hence knowing that there is a root in there. The example (and only example) that my professor gave us (and my professor said it would be just like this one) was this:

Show that there is a root of x⁵ - 2x⁴ - x - 3 = 0
between 2 and 3

f(2) ends up = -5 and f(3)= 75


Thanks again for your advice and taking the time to respond and explain.