a. Determine whether the Mean Value Theorem applies to the function on the given interval .
b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.
a. Choose the correct answer below.
A. The Mean Value Theorem applies because the function is continuous on and differentiable on .
B. The Mean Value Theorem does not apply because the function is not continuous on .
C. The Mean Value Theorem does not apply because the function is not differentiable on .
D. The Mean Value Theorem applies because the function is continuous on and differentiable on , In 7.
b. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The point(s) is/are .
(Type an exact answer. Use a comma to separate answers as needed.)
B. The Mean Value Theorem does not apply in this case.

Ask by Schneider Johnston. in the United States

Mar 29,2025

Question

a. Determine whether the Mean Value Theorem applies to the function on the given interval .
b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.
a. Choose the correct answer below.
A. The Mean Value Theorem applies because the function is continuous on and differentiable on .
B. The Mean Value Theorem does not apply because the function is not continuous on .
C. The Mean Value Theorem does not apply because the function is not differentiable on .
D. The Mean Value Theorem applies because the function is continuous on and differentiable on , In 7.
b. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The point(s) is/are .
(Type an exact answer. Use a comma to separate answers as needed.)
B. The Mean Value Theorem does not apply in this case.

Ask by Schneider Johnston. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

**Step 1. Verify the hypotheses of the Mean Value Theorem (MVT).**  
The MVT requires that the function be continuous on the closed interval and differentiable on the open interval.  
- The function $$ f(x)=e^{x} $$ is continuous for all $$ x \in \mathbb{R} $$, so it is continuous on $$ [0,\ln 7] $$.  
- The function is also differentiable for all $$ x \in \mathbb{R} $$, so it is differentiable on $$ (0,\ln 7) $$.

Thus, the MVT applies. The correct answer to part (a) is:  
**D.** The Mean Value Theorem applies because the function is continuous on $$ [0,\ln 7] $$ and differentiable on $$ (0, \ln 7) $$.

---

**Step 2. Find the point $$ c $$ that satisfies the conclusion of the MVT.**  
The Mean Value Theorem states that there exists at least one point $$ c $$ in $$ (0,\ln7) $$ such that

$$
f'(c) = \frac{f(\ln7)-f(0)}{\ln7-0}.
$$

1. **Compute $$ f(\ln7) $$ and $$ f(0) $$:**

$$
   f(\ln7)=e^{\ln7}=7 \quad \text{and} \quad f(0)=e^{0}=1.
   $$

2. **Calculate the slope:**

$$
   \frac{f(\ln7)-f(0)}{\ln7} = \frac{7-1}{\ln7} = \frac{6}{\ln7}.
   $$

3. **Differentiate $$ f(x) $$:**

$$
   f'(x)=e^{x}.
   $$

4. **Set up the equation:**

$$
   e^{c}=\frac{6}{\ln7}.
   $$

5. **Solve for $$ c $$:**

Taking the natural logarithm of both sides:

$$
   c=\ln\left(\frac{6}{\ln7}\right).
   $$

Thus, the point $$ c $$ is

$$
\boxed{c=\ln\left(\frac{6}{\ln7}\right)}.
$$

The correct answer for part (b) is:  
**A.** The point(s) is/are $$ x=\ln\left(\frac{6}{\ln7}\right) $$.
**Part (a):** The Mean Value Theorem applies because $$ f(x)=e^{x} $$ is continuous on $$ [0,\ln7] $$ and differentiable on $$ (0,\ln7) $$. **Part (b):** The point where the theorem applies is $$ x=\ln\left(\frac{6}{\ln7}\right) $$.

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