Show that there is exactly one real number for which . Find the integer part of this number. State any theorems you use in your solution.
By the intermediate value theorem, the continuous function crosses the x axis. The function is always increasing, therefore it crosses the x axis only once.
Is there an algebraic way for finding the integer part? I just subbed 1 into the original equation and got an answer < 5 then subbed in 2 and got an answer > 5 and deduced that x must be 1.something the integer part is 1.
which is positive for any value of x, therefore it is always increasing. Is that sufficient?
Originally I just noticed that the function was basically a shifted version of the e^x function and deemed it as always increasing Not very thorough.
he's left out some steps, but if you accept that the function is always increasing, then there must be exactly one zero.
@terrorsquid: it would be prudent to show that f'(x) > 0, for all x (EDIT: i didn't see your latest post).