For the given function, find all values of c over the interval (1,2) stipulated by Rolle's Theorem such that f' (c) = 0. x² f(x) = Зх — 2 | Submit multiple values as a comma-separated list. Do not round any value. c =

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question
### Finding Values of \( c \) Using Rolle's Theorem

For the given function, find all values of \( c \) over the interval \( (1,2) \) stipulated by Rolle's Theorem such that \( f'(c) = 0 \).

\[ f(x) = \frac{x^2}{3x - 2} \]

**Instructions:**
- Submit multiple values as a comma-separated list.
- Do not round any value.

**Solution:**

\[ c = \]

**Note:** To solve this problem, you need to apply Rolle's Theorem which requires:
1. \( f(x) \) is continuous on a closed interval \([a, b]\).
2. \( f(x) \) is differentiable on an open interval \( (a, b) \).
3. \( f(a) = f(b) \).

Once these conditions are verified, find the derivative \( f'(x) \) and solve for \( c \) in the given interval \( (1,2) \) where \( f'(c) = 0 \).
Transcribed Image Text:### Finding Values of \( c \) Using Rolle's Theorem For the given function, find all values of \( c \) over the interval \( (1,2) \) stipulated by Rolle's Theorem such that \( f'(c) = 0 \). \[ f(x) = \frac{x^2}{3x - 2} \] **Instructions:** - Submit multiple values as a comma-separated list. - Do not round any value. **Solution:** \[ c = \] **Note:** To solve this problem, you need to apply Rolle's Theorem which requires: 1. \( f(x) \) is continuous on a closed interval \([a, b]\). 2. \( f(x) \) is differentiable on an open interval \( (a, b) \). 3. \( f(a) = f(b) \). Once these conditions are verified, find the derivative \( f'(x) \) and solve for \( c \) in the given interval \( (1,2) \) where \( f'(c) = 0 \).
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