Does a Non-Injective Differentiable Function Have a Zero Derivative Point?

[2RIP]

If an interval is differentiable, but not injective, will there be a point where the derivative f'(x)=0 on that interval?

I'm not really sure how to approach this question. Help please?

 

[HallsofIvy]

Use the mean value theorem. f f(x) is not injective, then there must exist x1 and x2 such that f(x1)= f(x2). Apply the mean value theorem to the interval [x1, x2].

 

[Architeuthis Dux]

I understand your confusion. Let me explain the concept of injective functions and their relationship to derivatives.

A function is considered injective if each input has a unique output. In other words, no two inputs can have the same output. This is also known as the one-to-one mapping property. On the other hand, a function is non-injective if there exist two or more inputs that result in the same output.

When it comes to derivatives, they represent the rate of change of a function at a specific point. In other words, the derivative tells us how much the function is changing at a particular input. This means that for a function to have a derivative of zero at a certain point, the function must be either flat or have a constant slope at that point.

Now, let's apply this concept to a differentiable but non-injective function on an interval. Since the function is differentiable, it means that the derivative exists at every point on the interval. However, because the function is non-injective, there will be at least one point where two or more inputs result in the same output. This means that at that point, the derivative will be zero since the function is either flat or has a constant slope.

In conclusion, a non-injective function on an interval can have points where the derivative is zero, but this is not always the case. It ultimately depends on the specific function and its behavior on the interval. I hope this helps clarify the relationship between injective functions and derivatives.