Answer is: option2
\[ P'(a)(k - m) = P(k) - P(m) \]Solution:
The Mean Value Theorem (MVT) states that if a function \( P(x) \) is continuous on the closed interval \([k, m]\) and differentiable on the open interval \((k, m)\), then there exists a point \( a \) in \((k, m)\) such that:
\[ P'(a) = \frac{P(m) - P(k)}{m - k} \]
This can be rewritten as:
\[ P'(a)(m - k) = P(m) - P(k) \]
Now, let's analyze the given options:
Option A: \(\frac{P(k) - P(m)}{m - k} = P'(a)\)
This is equivalent to \(P'(a) = -\frac{P(m) - P(k)}{m - k}\), which is incorrect because the sign is wrong.
Option B: \(P'(a)(k - m) = P(k) - P(m)\)
This can be rewritten as \(P'(a) = \frac{P(k) - P(m)}{k - m} = \frac{P(m) - P(k)}{m - k}\), which matches the MVT. So, this is correct.
Option C: \(\frac{m - k}{P(m) - P(k)} = a\)
This is incorrect because the left-hand side is a ratio of differences, not equal to a point \(a\).
Option D: \(\frac{m - k}{P(m) - P(k)} = P'(a)\)
This is incorrect because it inverts the ratio compared to the MVT.
Final Answer: B