Q: Suppose that f: [0,2] -> [0,4] is continuous. Show that there is an x [0,2] such that .
Proof: All I know is use the intermediate value theorem...
That is the whole problem. But there is a hint here provided by my professor:
Hint: Consider the function g: [0,2] -> R defined by g(x) = f(x) - 2x and use the intermediate value theorem.
Now, again, we have not learn about using derivatives in proof yet.
Thank you!
Define a function on the interval as,
Since, are countinous on so too is continous on .
Now,
By definition
If then the proof is complete.
Because is one such point. So there is no harm in assuming that
By definition
If then the proof is complete.
Because is one such point. So there is no harm in assuming that
Thus, and .
Since is continous on there is a point such that by the intermediate value theorem.
Thus, . Thus, for some point in .
Q.E.D.