Using Lagrange's theorem , find the value of c for the following functions :
(i) `x^(3) - 3x^(2) + 2x` in the interval [0,1/2].
(ii) f(x) = `2x^(2) - 10x + 1` in the interval [2,7].
(iii) f(x) = (x-4) (x-6) in the interval [4,10].
(iv) f(x) = `sqrt(x-1)` in the interval [1,3].
(v) f(x) = `2x^(2) + 3x + 4` in the interval [1,2].

Answer

To solve the problem using Lagrange's Mean Value Theorem (LMVT), we need to follow these steps for each function provided. The theorem states that if a function \( f(x) \) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \( c \) in \((a, b)\) such that:

\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

Let's solve each part step by step.

### (i) \( f(x) = x^3 - 3x^2 + 2x \) in the interval \([0, \frac{1}{2}]\)

1. **Check continuity and differentiability**:
- \( f(x) \) is a polynomial function, hence continuous and differentiable everywhere.

2. **Calculate \( f(0) \) and \( f\left(\frac{1}{2}\right) \)**:
- \( f(0) = 0^3 - 3(0^2) + 2(0) = 0 \)
- \( f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 - 3\left(\frac{1}{2}\right)^2 + 2\left(\frac{1}{2}\right) = \frac{1}{8} - \frac{3}{4} + 1 = \frac{1}{8} - \frac{6}{8} + \frac{8}{8} = \frac{3}{8} \)

3. **Apply LMVT**:
- Calculate the slope:
\[ \frac{f\left(\frac{1}{2}\right) - f(0)}{\frac{1}{2} - 0} = \frac{\frac{3}{8} - 0}{\frac{1}{2}} = \frac{3/8}{1/2} = \frac{3}{4} \]

4. **Find \( c \)**:
- Differentiate \( f(x) \):
\[ f'(x) = 3x^2 - 6x + 2 \]
- Set \( f'(c) = \frac{3}{4} \):
\[ 3c^2 - 6c + 2 = \frac{3}{4} \]
- Multiply through by 4 to eliminate the fraction:
\[ 12c^2 - 24c + 8 = 3 \implies 12c^2 - 24c + 5 = 0 \]

5. **Solve the quadratic equation**:
\[ c = \frac{24 \pm \sqrt{(-24)^2 - 4 \cdot 12 \cdot 5}}{2 \cdot 12} = \frac{24 \pm \sqrt{576 - 240}}{24} = \frac{24 \pm \sqrt{336}}{24} = \frac{24 \pm 4\sqrt{21}}{24} = 1 \pm \frac{\sqrt{21}}{6} \]
- The two values for \( c \) are \( 1 + \frac{\sqrt{21}}{6} \) and \( 1 - \frac{\sqrt{21}}{6} \).

6. **Check which value lies in \( (0, \frac{1}{2}) \)**:
- \( 1 - \frac{\sqrt{21}}{6} \approx 0.236 \) (acceptable)
- \( 1 + \frac{\sqrt{21}}{6} \approx 2.87 \) (not acceptable)