Whether the statement “There exists a function f such that and for all x” is true or false.
The given statement is false.
Mean value theorem:
If f is a continuous function on such that f is differentiable on , then there exits a point c in such that
Given statement is false since the following example disproves it.
Consider the two points .
The slope of the line joining these points is given by
Note that the slope represents the first derivative of the function.
Thus, there exists some point c in for which the value of first derivative is 1.
Therefore for all x is not possible.
Therefore, the given statement is false.
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