Question

Let \( f'(x) \) be differentiable for all \( x \). If \( f(1) = -2 \) and \[ f'(x) \geq 2 \quad \forall x \in [1, 6], \] then:

• A. \( f(6)<8 \)
• B. \( f(6) \geq 8 \)
• C. \( f(6) \geq 5 \)
• D. \( f(6) \leq 5 \)

Hint:

When applying the Mean Value Theorem, the value of the function at the endpoints and the derivative must be considered to derive an inequality.

Answer 1

The Correct Option is B

Step 1: Applying the Mean Value Theorem.
By the Mean Value Theorem, since \( f(x) \) is differentiable on the interval \( [1, 6] \), there exists a point \( c \in (1, 6) \) such that: \[ f'(c) = \frac{f(6) - f(1)}{6 - 1}. \] We know that \( f(1) = -2 \) and \( f'(x) \geq 2 \) for all \( x \in [1, 6] \).
Step 2: Substituting known values.
Substituting the known values into the Mean Value Theorem equation, we get: \[ f'(c) = \frac{f(6) - (-2)}{5}. \] Since \( f'(x) \geq 2 \), we have: \[ \frac{f(6) + 2}{5} \geq 2. \] Multiplying both sides by 5: \[ f(6) + 2 \geq 10. \] Thus: \[ f(6) \geq 8. \]
Conclusion.
The correct answer is (2) \( f(6) \geq 8 \), as we have shown that \( f(6) \) must be at least 8.