Question: User: Question 20
For f(x) = \{x + 1, x \geq 0
\{e^x, x < 0 f'(x) =
A. \{x, x \geq 0
\{e^x, x < 0
B. \{1, x \geq 0
\{e^x, x < 0
C. \{x, x > 0
\{e^x, x < 0
D. \{1, x > 0
\{e^x, x < 0
E. \{1, e^x > 0
\{e^x, e^x < 0
Question 21
For f(x) = \{x + 1, x \geq 0
\{e^x, x < 0 f'(-1) =
A. \{0, x \geq 0
\{e^{-1}, x < 0
B. \{1, x \geq 0
\{e^{-1}, x < 0
C. e^{-1}
D. 0
E. 1
Question 22
P(x) = ax^3 + bx^2 - cx - 1 is a cubic polynomial and has factors of (x - 1) and (x + 3). The function
f(x) = ax^3 + bx^2 - cx - 1 has a stationary point at x = 0. The values of a, b and c are
A. a = 0, b = \frac{5}{9}, c = \frac{7}{9}
B. a = \frac{2}{9}, b = \frac{7}{9}, c = 0
C. a = \frac{2}{9}, b = \frac{7}{9}, c = c
D. no solution possible
E. a = 1, b = -2, c = 0
Question 23
f(x) = 3 \cos(x) and g(x) = 2 \log_e(x) for maximal domains, the function f(g(x)) can be defined as
A. f(g(x)) = 3 \cos(2 \log_e(x)), x \in \mathbb{R}
B. f(g(x)) = 2 \log_e(3 \cos(x)), x \in \mathbb{R}
C. f(g(x)) = 3 \cos(2 \log_e(x)), x \in (0, \infty)
D. f(g(x)) = 2 \log_e(\cos(x)) + 2 \log_e(3), x \in [-1, 1]
E. f(g(x)) = 2 \log_e(\cos(x)) + 2 \log_e(3), x \in [0, \infty)
1. **Problem Statement:** Find the derivative $f'(x)$ of the piecewise function
$$f(x) = \begin{cases} x + 1, & x \geq 0 \\ e^x, & x < 0 \end{cases}$$
2. **Formula and Rules:** The derivative of $x + 1$ is $1$ and the derivative of $e^x$ is $e^x$.
3. **Derivative for each piece:**
- For $x \geq 0$, $f'(x) = 1$
- For $x < 0$, $f'(x) = e^x$
4. **Check continuity at $x=0$:**
- Left-hand derivative at $0$ is $e^0 = 1$
- Right-hand derivative at $0$ is $1$
5. **Conclusion:** The derivative is
$$f'(x) = \begin{cases} 1, & x \geq 0 \\ e^x, & x < 0 \end{cases}$$
6. **Answer:** Option B matches this derivative.
**Final answer:** B