Verify cmvt for f(x)=3x+2, g(x)=x^2+1 on [1,4]

To verify the Cauchy Mean Value Theorem (CMVT) for the functions \( f(x) = 3x + 2 \) and \( g(x) = x^2 + 1 \) on the interval \([1, 4]\), we follow these steps:

1. **State the Cauchy Mean Value Theorem:**

   The CMVT states that if \( f \) and \( g \) are continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists at least one \( c \in (a, b) \) such that:
   \[
   \frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}
   \]

2. **Check the conditions for CMVT:**

   - Both \( f(x) = 3x + 2 \) and \( g(x) = x^2 + 1 \) are continuous and differentiable on \([1, 4]\).
   - Therefore, the conditions for the CMVT are satisfied.

3. **Compute \( f(a) \), \( f(b) \), \( g(a) \), and \( g(b) \):**

   \[
   f(1) = 3 \cdot 1 + 2 = 5
   \]
   \[
   f(4) = 3 \cdot 4 + 2 = 14
   \]
   \[
   g(1) = 1^2 + 1 = 2
   \]
   \[
   g(4) = 4^2 + 1 = 17
   \]

4. **Compute the difference quotient:**

   \[
   \frac{f(4) - f(1)}{g(4) - g(1)} = \frac{14 - 5}{17 - 2} = \frac{9}{15} = \frac{3}{5}
   \]

5. **Compute the derivatives \( f'(x) \) and \( g'(x) \):**

   \[
   f'(x) = 3
   \]
   \[
   g'(x) = 2x
   \]

6. **Set up the CMVT equation:**

   According to the CMVT, there exists a \( c \in (1, 4) \) such that:
   \[
   \frac{f'(c)}{g'(c)} = \frac{3}{2c} = \frac{3}{5}
   \]

7. **Solve for \( c \):**

   \[
   \frac{3}{2c} = \frac{3}{5}
   \]
   \[
   3 \cdot 5 = 3 \cdot 2c
   \]
   \[
   15 = 6c
   \]
   \[
   c = \frac{15}{6} = 2.5
   \]

8. **Conclusion:**

   There exists \( c = 2.5 \in (1, 4) \) such that:
   \[
   \frac{f'(c)}{g'(c)} = \frac{f(4) - f(1)}{g(4) - g(1)}
   \]

Thus, the Cauchy Mean Value Theorem is verified for \( f(x) = 3x + 2 \) and \( g(x) = x^2 + 1 \) on the interval \([1, 4]\).

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What is the significance of the Cauchy Mean Value Theorem in calculus?
2. How does the Cauchy Mean Value Theorem differ from the Mean Value Theorem?
3. Can the Cauchy Mean Value Theorem be applied to non-polynomial functions?
4. What are the conditions under which the Cauchy Mean Value Theorem can be applied?
5. How can we interpret the result of the Cauchy Mean Value Theorem geometrically?
6. Are there any practical applications of the Cauchy Mean Value Theorem?
7. How does the Cauchy Mean Value Theorem relate to the concept of derivatives?
8. Can we use the Cauchy Mean Value Theorem to solve real-world problems in physics or engineering?

**Tip:** When verifying theorems like the CMVT, carefully check that all conditions are met before proceeding with the calculations.

Learn how to verify the Cauchy Mean Value Theorem (CMVT) for the functions f(x) = 3x + 2 and g(x) = x^2 + 1 on the interval [1, 4]. Follow a detailed step-by-step solution involving differentiation, computing function values, and applying the CMVT theorem. Suitable for advanced high school students and those studying calculus.

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