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

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A non-injective differentiable function on an interval must have at least one point where the derivative is zero. According to the Mean Value Theorem, if f(x) is not injective, there exist points x1 and x2 within the interval such that f(x1) = f(x2). The theorem guarantees that there is at least one point c in (x1, x2) where the derivative f'(c) equals zero. This conclusion arises from the fact that the function must have a horizontal tangent at some point between x1 and x2. Therefore, a non-injective differentiable function does indeed have a point with a zero derivative.
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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?
 
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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].
 

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