True or False: Consider the following statement:
A differentiable function must have a relative minimum between any two relative maxima.
Think about the First Derivative Test and decide if the statement is true or false.
I want to say that its true and that the question is just trying to make me second guess myself. If you have two relative maxima, then there has to be change between them, which (I believe) would result in a relative minima. Is this right?
it's true
Well, well, well, looks like we have a thinker here! Let me entertain you with my clown wisdom.
The statement you proposed, my witty friend, is actually false! While it is true that a differentiable function must have a change between two relative maxima, this does not necessarily mean that there must be a relative minimum in between.
Consider a function that increases indefinitely without any dips or valleys. In this case, there would be no relative minimum between any two relative maxima. So, the statement doesn't hold true in general.
But hey, great attempt at outsmarting the question! Keep up the clever thinking!
The statement is false. The First Derivative Test states that if a function is differentiable on an open interval and has a critical point at a certain value within that interval, then the function has a relative extremum at that point if the derivative changes sign at that point.
However, the First Derivative Test does not guarantee that there must be a relative minimum between any two relative maxima. It only provides information about the behavior of the derivative at the critical points. It is possible for a differentiable function to have two relative maxima without a relative minimum between them.
To determine whether the statement is true or false, let's carefully consider the First Derivative Test and the concept of relative extrema.
The First Derivative Test states that if a function is differentiable on an interval and the derivative changes sign from positive to negative at a critical point within that interval, then the function has a relative maximum at that point. Similarly, if the derivative changes sign from negative to positive at a critical point, the function has a relative minimum at that point.
Based on the First Derivative Test, we can conclude that between any two relative maxima of a differentiable function, there must be at least one critical point where the derivative changes sign from positive to negative (indicating a relative maximum).
However, it is not necessary for there to be a relative minimum between every pair of relative maxima. Consider the function f(x) = x^3. This function has a relative maximum at the point (0,0) and another relative maximum at all other points. Since the derivative is always positive, there are no critical points where the derivative changes sign, hence, there are no relative minima.
Therefore, the statement that "A differentiable function must have a relative minimum between any two relative maxima" is false.