Question

If a function \(f:[0,1]\to\mathbb{R}\) is differentiable, \(f'(0) = -1\) and \(f'(1) = 5\), then which one of the following statements is true?

• A. There exists a point \(c\in(0,1)\) such that \(f'(c) = -1\).
• B. f' must be continuous on [0,1].
• C. There exists a point \(c\in(0,1)\) such that \(f'(c) = 4\).
• D. f' must be differentiable on (0,1).

Hint:

Apply Darboux's theorem, which gives derivatives the intermediate value property even when the derivative is not continuous.

Answer 1

The Correct Option is C

Step 1: Recall Darboux's theorem. If \(f\) is differentiable on \([a,b]\), then \(f'\) has the intermediate value property on \([a,b]\): for any value \(k\) strictly between \(f'(a)\) and \(f'(b)\), there exists \(c \in (a,b)\) with \(f'(c) = k\). This holds even if \(f'\) is not continuous, unlike the ordinary intermediate value theorem which requires continuity.

Step 2: Check option (C) against Darboux's theorem. Here \(f'(0) = -1\) and \(f'(1) = 5\). The value \(k = 4\) lies strictly between \(-1\) and \(5\). By Darboux's theorem, there must exist some \(c \in (0,1)\) with \(f'(c) = 4\). So option (C) is guaranteed true.

Step 3: Rule out option (A). Darboux's theorem guarantees intermediate values are attained somewhere in \((0,1)\), but \(-1\) is an endpoint value, not necessarily attained again in the open interval. For example, if \(f'\) increases strictly and monotonically from \(-1\) to \(5\) on \([0,1]\), it never returns to \(-1\) inside \((0,1)\). So (A) is not guaranteed.

Step 4: Rule out option (B). Derivatives satisfy the Darboux (intermediate value) property, but they need not be continuous. A classical example is \(g(x) = x^2\sin(1/x)\) for \(x\neq0\), \(g(0)=0\), whose derivative exists everywhere but is discontinuous at \(0\). So differentiability of \(f\) does not force continuity of \(f'\); (B) is false in general.

Step 5: Rule out option (D). There is no theorem requiring the derivative of a differentiable function to itself be differentiable; \(f'\) can fail to be differentiable, or even fail to be continuous, at various points. So (D) is not guaranteed.

Step 6: Conclude. Only option (C) is a valid consequence, directly from Darboux's theorem. \[\boxed{\text{There exists } c\in(0,1) \text{ such that } f'(c)=4}\]