Mean Value Theorem

The Lagrange's Mean Value Theorem (LMVT) states that if a function is continuous on the closed interval [ ] and differentiable on the open interval ( ), there exists at least one in the interval ( ) such that:

Let's apply this theorem to the function with and .

1. First, check the conditions of the theorem:
   • Continuity: The function is continuous for because the expression under the square root is positive or zero (i.e., ).
   • Differentiability: The function is differentiable for . Thus, it satisfies all conditions of LMVT on this interval.
2. Calculate and :
3. Find the derivative :
   • Using the chain rule,
4. Apply LMVT:

According to LMVT, there exists some in (2, 3) such that:

• This becomes
• Simplifying gives:
• Let us solve for :
  • Cross-multiply to get
  • Square both sides:
  • Expanding:
  • Rearrange to find: (Only consider positive because is in (2, 3))

1. Conclusion: The value of prescribed by the Lagrange's Mean Value Theorem is .