Problem 8: Let $$ f(x)=3 x^{3}-2 x^{2}+x $$ on the interval $$ [-1,2] $$. . Apply the MVT to find the value(s) of $$ c $$ in $$ (-1,2) $$ where $$ f^{\prime}(c)=\frac{f(2)-f(-1)}{2-(-1)} $$.

Question

UpStudy AI · Accepted Answer

To apply the Mean Value Theorem (MVT) to the function $$ f(x) = 3x^3 - 2x^2 + x $$ on the interval $$[-1, 2]$$, follow these steps:

### **1. Verify the Conditions for MVT**

- **Continuity:** $$ f(x) $$ is a polynomial, so it is continuous on $$[-1, 2]$$.
- **Differentiability:** Since $$ f(x) $$ is a polynomial, it is differentiable on $$(-1, 2)$$.

### **2. Apply the Mean Value Theorem**

According to the MVT, there exists at least one $$ c $$ in $$(-1, 2)$$ such that:
$$
f'(c) = \frac{f(2) - f(-1)}{2 - (-1)}
$$

### **3. Compute $$ f(2) $$ and $$ f(-1) $$**

$$
f(2) = 3(2)^3 - 2(2)^2 + 2 = 24 - 8 + 2 = 18
$$
$$
f(-1) = 3(-1)^3 - 2(-1)^2 + (-1) = -3 - 2 - 1 = -6
$$

### **4. Calculate the Average Rate of Change**

$$
\frac{f(2) - f(-1)}{2 - (-1)} = \frac{18 - (-6)}{3} = \frac{24}{3} = 8
$$

### **5. Find $$ f'(x) $$ and Solve for $$ c $$**

First, find the derivative $$ f'(x) $$:
$$
f'(x) = 9x^2 - 4x + 1
$$
Set $$ f'(c) = 8 $$:
$$
9c^2 - 4c + 1 = 8 \
9c^2 - 4c - 7 = 0
$$

### **6. Solve the Quadratic Equation**

$$
c = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 9 \cdot (-7)}}{2 \cdot 9} = \frac{4 \pm \sqrt{16 + 252}}{18} = \frac{4 \pm \sqrt{268}}{18}
$$
Simplify the square root:
$$
\sqrt{268} = 2\sqrt{67}
$$
Thus,
$$
c = \frac{4 \pm 2\sqrt{67}}{18} = \frac{2 \pm \sqrt{67}}{9}
$$

### **7. Verify $$ c $$ is in $$(-1, 2)$$**

Calculate approximate values:
$$
\sqrt{67} \approx 8.185
$$
$$
c_1 = \frac{2 + 8.185}{9} \approx 1.132 \quad (\text{within } (-1, 2))
$$
$$
c_2 = \frac{2 - 8.185}{9} \approx -0.687 \quad (\text{within } (-1, 2))
$$

Both values of $$ c $$ lie within the interval $$(-1, 2)$$.

### **Final Answer**

The values of $$ c $$ in $$(-1, 2)$$ satisfying the Mean Value Theorem are:
$$
c = \frac{2 + \sqrt{67}}{9} \quad \text{and} \quad c = \frac{2 - \sqrt{67}}{9}
$$

**Answer:** After simplifying, the values c are (2 ± √67) divided by 9. Thus,

c = (2 ± √67)/9
The values of $$ c $$ in $$(-1, 2)$$ that satisfy the Mean Value Theorem are $$ c = \frac{2 + \sqrt{67}}{9} $$ and $$ c = \frac{2 - \sqrt{67}}{9} $$.

Problem 8:
Let on the interval .
. Apply the MVT to find the value(s) of in where .

Upstudy AI Solution

Answer

Solution

1. Verify the Conditions for MVT

2. Apply the Mean Value Theorem

3. Compute and

4. Calculate the Average Rate of Change

5. Find and Solve for

6. Solve the Quadratic Equation

7. Verify is in

Final Answer

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Problem 8: Let on the interval . . Apply the MVT to find the value(s) of in where .