Let f be a function that is differentiable on the open interval (1,10) . If f(2)=-5, f(5)=5, and f(9)=-5, which of the
following must be true?
I. f has at least 2 zeros.
II. The graph of f has at least one horizontal tangent.
III. For some c, 2<c<5, f(c)=3.
please explain how statement II is true?
To determine whether statement II is true or false, we need to look at the behavior of the derivative of f on the interval (1,10).
Since f is differentiable on (1,10), it means that its derivative exists and is continuous on the same interval. According to the Mean Value Theorem, there exists at least one point c in the interval (1,10) such that:
f'(c) = (f(10) - f(1))/(10 - 1)
We know that f(2) = -5 and f(9) = -5. So, we can rewrite the above equation as:
f'(c) = (-5 - (-5))/(9 - 2) = 0
This means that at some point c in the interval (1,10), the derivative of f is equal to zero. Geometrically, this corresponds to a horizontal tangent line on the graph of f at the point (c, f(c)).
Therefore, statement II is true.