Verify Lagrange mean value for f(x) = x ^ 3 + x ^ 2 in [1,2]

To verify the Lagrange Mean Value Theorem (LMVT) for the function \( f(x) = x^3 + x^2 \) on the interval \([1, 2]\), let's proceed step by step.

### Step 1: Verify Conditions of LMVT
LMVT states that if a function \( f(x) \) is:
1. Continuous on a closed interval \([a, b]\),
2. Differentiable on an open interval \((a, b)\),

then there exists at least one point \( c \in (a, b) \) such that:
\[
f'(c) = \frac{f(b) - f(a)}{b - a}
\]

The function \( f(x) = x^3 + x^2 \) is a polynomial, which is continuous and differentiable everywhere, so it satisfies both conditions on \([1, 2]\).

### Step 2: Calculate \( f(1) \) and \( f(2) \)
Let's find \( f(1) \) and \( f(2) \) to compute the slope of the secant line between these points:
\[
f(1) = 1^3 + 1^2 = 1 + 1 = 2
\]
\[
f(2) = 2^3 + 2^2 = 8 + 4 = 12
\]

### Step 3: Calculate the Secant Slope
The secant slope between \( x = 1 \) and \( x = 2 \) is:
\[
\frac{f(2) - f(1)}{2 - 1} = \frac{12 - 2}{2 - 1} = \frac{10}{1} = 10
\]

### Step 4: Find \( f'(x) \)
Now, we calculate the derivative \( f'(x) \):
\[
f(x) = x^3 + x^2
\]
\[
f'(x) = 3x^2 + 2x
\]

### Step 5: Set \( f'(c) = 10 \) and Solve for \( c \)
We now need to find \( c \in (1, 2) \) such that \( f'(c) = 10 \):
\[
3c^2 + 2c = 10
\]
Rearrange to form a standard quadratic equation:
\[
3c^2 + 2c - 10 = 0
\]

Solve this quadratic equation using the quadratic formula:
\[
c = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 3 \cdot (-10)}}{2 \cdot 3}
\]
\[
c = \frac{-2 \pm \sqrt{4 + 120}}{6}
\]
\[
c = \frac{-2 \pm \sqrt{124}}{6}
\]
\[
c = \frac{-2 \pm 2\sqrt{31}}{6}
\]
\[
c = \frac{-1 \pm \sqrt{31}}{3}
\]

Calculating these values:
\[
c \approx 1.62 \quad \text{(since \( 1 < c < 2 \))}
\]

### Conclusion
There exists a point \( c \approx 1.62 \) in the interval \((1, 2)\) where \( f'(c) = 10 \), thus satisfying the Lagrange Mean Value Theorem.

Would you like more details or have any questions?

Here are some related questions:
1. How does the Mean Value Theorem differ from Rolle's Theorem?
2. Can LMVT be applied to trigonometric functions over specified intervals?
3. What are the implications of LMVT in real-world scenarios?
4. How does LMVT apply to non-polynomial functions?
5. How can we approximate values using LMVT in calculus problems?

**Tip:** Always check the continuity and differentiability of the function over the interval to verify conditions for applying LMVT.

Step-by-step solution to verify the Lagrange Mean Value Theorem for the function f(x) = x^3 + x^2 on the interval [1, 2]. Learn how to apply the theorem, compute derivatives, and solve for the point c where f'(c) equals the secant slope.

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