A smooth journey has an instant whose speed matches its average speed.
The Mean Value Theorem assumes continuity on [a,b] and differentiability on (a,b). For f(x)=x², the endpoint slope is a+b. Setting 2c=a+b gives c=(a+b)/2, which lies strictly inside a nonzero interval. Other functions can have more than one such c.
The theorem requires its continuity and differentiability hypotheses; endpoint agreement alone is not enough.