13. Verify that the hypotheses of the Mean Value Theorem are satisfied on the given interval, and find all values of $$ c $$ in that interval that satisfy the conclusion of the theorem. a) $$ f(x)=x^{2}+x ;[-4,6] $$ d) $$ f(x)=\sqrt{2}-x-x^{2} $$; b) $$ \left.\left.f(x)=x^{3}+x-4 ;-1,2\right]\right] $$ c) $$ f(x)=x+\frac{1}{x} $$; 3,4$$ ] $$

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To verify the hypotheses of the Mean Value Theorem (MVT) and find the values of $$ c $$ that satisfy the conclusion, we need to follow these steps for each function:

1. **Check if the function is continuous on the closed interval $$[a, b]$$**.
2. **Check if the function is differentiable on the open interval $$(a, b)$$**.
3. **Calculate the average rate of change of the function over the interval**.
4. **Find $$ c $$ such that $$ f'(c) $$ equals the average rate of change**.

Let's go through each function step by step.

### a) $$ f(x) = x^2 + x $$ on $$[-4, 6]$$

1. **Continuity**: The function $$ f(x) = x^2 + x $$ is a polynomial, hence continuous on $$[-4, 6]$$.
2. **Differentiability**: Polynomials are also differentiable everywhere, so $$ f(x) $$ is differentiable on $$(-4, 6)$$.
3. **Average Rate of Change**:
   $$
   \text{Average Rate} = \frac{f(6) - f(-4)}{6 - (-4)} = \frac{(6^2 + 6) - ((-4)^2 + (-4))}{10} = \frac{(36 + 6) - (16 - 4)}{10} = \frac{42 - 12}{10} = \frac{30}{10} = 3
   $$
4. **Find $$ c $$**:
   $$
   f'(x) = 2x + 1
   $$
   Set $$ f'(c) = 3 $$:
   $$
   2c + 1 = 3 \implies 2c = 2 \implies c = 1
   $$

### b) $$ f(x) = x^3 + x - 4 $$ on $$[-1, 2]$$

1. **Continuity**: The function $$ f(x) = x^3 + x - 4 $$ is a polynomial, hence continuous on $$[-1, 2]$$.
2. **Differentiability**: Polynomials are also differentiable everywhere, so $$ f(x) $$ is differentiable on $$(-1, 2)$$.
3. **Average Rate of Change**:
   $$
   \text{Average Rate} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{(2^3 + 2 - 4) - ((-1)^3 + (-1) - 4)}{3} = \frac{(8 + 2 - 4) - (-1 - 1 - 4)}{3} = \frac{6 - (-6)}{3} = \frac{12}{3} = 4
   $$
4. **Find $$ c $$**:
   $$
   f'(x) = 3x^2 + 1
   $$
   Set $$ f'(c) = 4 $$:
   $$
   3c^2 + 1 = 4 \implies 3c^2 = 3 \implies c^2 = 1 \implies c = 1 \text{ or } c = -1
   $$
   Only $$ c = 1 $$ is in the interval $$(-1, 2)$$.

### c) $$ f(x) = x + \frac{1}{x} $$ on $$ (3, 4) $$

1. **Continuity**: The function $$ f(x) = x + \frac{1}{x} $$ is continuous on $$ (3, 4) $$ since $$ x \neq 0 $$.
2. **Differentiability**: The function is differentiable on $$ (3, 4) $$ since it is defined and smooth in that interval.
3. **Average Rate of Change**:
   $$
   \text{Average Rate} = \frac{f(4) - f(3)}{4 - 3} = f(4) - f(3) = \left(4 + \frac{1}{4}\right) - \left(3 + \frac{1}{3}\right) = \left(4.25 - 3.3333\right) = 0.9167
   $$
4. **Find $$ c $$**:
   $$
   f'(x) = 1 - \frac{1}{x^2}
   $$
   Set $$ f'(c) = 0.9167 $$:
   $$
   1 - \frac{1}{c^2} = 0.9167 \implies \frac{1}{c^2} = 0.0833 \implies c^2 = 12 \implies c = \sqrt{12} \approx 3.464
   $$
   This value is in the interval $$ (3, 4) $$.

### d) $$ f(x) = \sqrt{2} - x - x^2 $$

1. **Continuity**: The function $$ f(x) = \sqrt{2} - x - x^2 $$ is a polynomial, hence continuous on any interval.
2. **Differentiability**: Polynomials are also differentiable everywhere.
3. **Average Rate of Change**: We need the interval to calculate this. Assuming the interval is $$[-a, b]$$ (please provide the interval).
4. **Find $$ c $$**: This will depend on the average rate of change calculated in the previous step.

### Summary of Results
- For $$ f(x) = x^2 + x $$, $$ c = 1 $$.
- For $$ f(x) = x^3 + x - 4 $$, $$ c = 1 $$.
- For $$ f(x) = x + \frac{1}{x} $$, $$ c \approx 3.464 $$.
- For $$ f(x) = \sqrt{2} - x - x^2 $$, please provide the interval to proceed.
For each function, the Mean Value Theorem is satisfied, and the values of $$ c $$ that satisfy the theorem are: - **a)** $$ c = 1 $$ - **b)** $$ c = 1 $$ - **c)** $$ c \approx 3.464 $$ - **d)** (No interval provided)

Verify that the hypotheses of the Mean Value Theorem are satisfied on the given interval, and find all values
of in that interval that satisfy the conclusion of the theorem.
a)
d) ;
b)
c) ; 3,4

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Verify that the hypotheses of the Mean Value Theorem are satisfied on the given interval, and find all values of in that interval that satisfy the conclusion of the theorem. a) d) ; b) c) ; 3,4