(29) 1. The critical numbers of $$ g(x)=\sin x+\cos x $$ on the open interval $$ (0,2 \pi) $$ are (A) $$ \frac{\pi}{4} $$ (B) $$ \frac{3 \pi}{4} $$ and $$ \frac{7 \pi}{4} $$ (C) $$ \frac{\pi}{4} $$ and $$ \frac{5 \pi}{4} $$ (D) $$ \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4} $$ and $$ \frac{7 \pi}{4} $$ 2. On the closed interval $$ [0,2 \pi] $$, the absolute minimum of $$ f(x)=e^{\sin x} $$ occurs at (A) 0 (B) $$ \frac{\pi}{2} $$ (C) $$ \frac{3 \pi}{2} $$ (D) $$ 2 \pi $$ The ranximum value of $$ f(x)=2 x^{3}-15 x^{2}+36 x $$ on the closed interval $$ [0,4] $$ is (A) 28 (Ti) 30 (C) 32 (C) 48 4. On the closed irturval $$ [0.5] $$, the function $$ f(x)=3-4 $$ - (A) both an at soly $$ { }^{\prime} $$ o maximum and an absolute minimen (B) an absoluse tuaximum but no absolute minimum. (C) no absoluts maximum but an absolute minimum. (D) un absoluie riarixaum and two absolute minima. 5. The critical numocrs of the function $$ f(x)=\left\{\begin{array}{ll}x^{2}+1 & ext { if }-2 \leq x \leq 1 \ 3 x^{2}-4 x+3 & ext { if } 1

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(29) 1. The critical numbers of $$ g(x)=\sin x+\cos x $$ on the open interval $$ (0,2 \pi) $$ are (A) $$ \frac{\pi}{4} $$ (B) $$ \frac{3 \pi}{4} $$ and $$ \frac{7 \pi}{4} $$ (C) $$ \frac{\pi}{4} $$ and $$ \frac{5 \pi}{4} $$ (D) $$ \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4} $$ and $$ \frac{7 \pi}{4} $$ 2. On the closed interval $$ [0,2 \pi] $$, the absolute minimum of $$ f(x)=e^{\sin x} $$ occurs at (A) 0 (B) $$ \frac{\pi}{2} $$ (C) $$ \frac{3 \pi}{2} $$ (D) $$ 2 \pi $$ The ranximum value of $$ f(x)=2 x^{3}-15 x^{2}+36 x $$ on the closed interval $$ [0,4] $$ is (A) 28 (Ti) 30 (C) 32 (C) 48 4. On the closed irturval $$ [0.5] $$, the function $$ f(x)=3-4 $$ - (A) both an at soly $$ { }^{\prime} $$ o maximum and an absolute minimen (B) an absoluse tuaximum but no absolute minimum. (C) no absoluts maximum but an absolute minimum. (D) un absoluie riarixaum and two absolute minima. 5. The critical numocrs of the function $$ f(x)=\left\{\begin{array}{ll}x^{2}+1 & 	ext { if }-2 \leq x \leq 1 \ 3 x^{2}-4 x+3 & 	ext { if } 1

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Let's solve each of the problems step by step.

### Problem 1: Critical Numbers of $$ g(x) = \sin x + \cos x $$

To find the critical numbers, we need to find the derivative of $$ g(x) $$ and set it to zero.

1. **Find the derivative:**
   $$
   g'(x) = \cos x - \sin x
   $$

2. **Set the derivative to zero:**
   $$
   \cos x - \sin x = 0
   $$
   This implies:
   $$
   \cos x = \sin x
   $$

3. **Solve for $$ x $$:**
   The solutions to $$ \cos x = \sin x $$ occur at:
   $$
   x = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z})
   $$
   Within the interval $$ (0, 2\pi) $$, the solutions are:
   $$
   x = \frac{\pi}{4} \quad \text{and} \quad x = \frac{5\pi}{4}
   $$

Thus, the critical numbers are $$ \frac{\pi}{4} $$ and $$ \frac{5\pi}{4} $$. The correct answer is:
**(C) $$ \frac{\pi}{4} $$ and $$ \frac{5\pi}{4} $$**

---

### Problem 2: Absolute Minimum of $$ f(x) = e^{\sin x} $$ on $$ [0, 2\pi] $$

1. **Find the derivative:**
   $$
   f'(x) = e^{\sin x} \cos x
   $$

2. **Set the derivative to zero:**
   $$
   e^{\sin x} \cos x = 0
   $$
   Since $$ e^{\sin x} $$ is never zero, we have:
   $$
   \cos x = 0
   $$
   This occurs at:
   $$
   x = \frac{\pi}{2} \quad \text{and} \quad x = \frac{3\pi}{2}
   $$

3. **Evaluate $$ f(x) $$ at critical points and endpoints:**
   - $$ f(0) = e^{\sin(0)} = e^0 = 1 $$
   - $$ f\left(\frac{\pi}{2}\right) = e^{\sin\left(\frac{\pi}{2}\right)} = e^1 = e $$
   - $$ f\left(\frac{3\pi}{2}\right) = e^{\sin\left(\frac{3\pi}{2}\right)} = e^{-1} = \frac{1}{e} $$
   - $$ f(2\pi) = e^{\sin(2\pi)} = e^0 = 1 $$

The minimum value occurs at $$ x = \frac{3\pi}{2} $$ with $$ f\left(\frac{3\pi}{2}\right) = \frac{1}{e} $$. The correct answer is:
**(C) $$ \frac{3\pi}{2} $$**

---

### Problem 3: Maximum Value of $$ f(x) = 2x^3 - 15x^2 + 36x $$ on $$ [0, 4] $$

1. **Find the derivative:**
   $$
   f'(x) = 6x^2 - 30x + 36
   $$

2. **Set the derivative to zero:**
   $$
   6x^2 - 30x + 36 = 0
   $$
   Dividing by 6:
   $$
   x^2 - 5x + 6 = 0
   $$
   Factoring:
   $$
   (x - 2)(x - 3) = 0
   $$
   Thus, $$ x = 2 $$ and $$ x = 3 $$.

3. **Evaluate $$ f(x) $$ at critical points and endpoints:**
   - $$ f(0) = 0 $$
   - $$ f(2) = 2(2^3) - 15(2^2) + 36(2) = 16 - 60 + 72 = 28 $$
   - $$ f(3) = 2(3^3) - 15(3^2) + 36(3) = 54 - 135 + 108 = 27 $$
   - $$ f(4) = 2(4^3) - 15(4^2) + 36(4) = 128 - 240 + 144 = 32 $$

The maximum value occurs at $$ x = 4 $$ with $$ f(4) = 32 $$. The correct answer is:
**(C) 32**

---

### Problem 4: Behavior of $$ f(x) = 3 - 4x $$ on $$ [0, 1] $$

1. **Evaluate $$ f(x) $$ at endpoints:**
   - $$ f(0) = 3 - 4(0) = 3 $$
   - $$ f(1) = 3 - 4(1) = -1 $$

Since $$ f(x) $$ is a linear function, it will have an absolute maximum at $$ x = 0 $$ and an absolute minimum at $$ x = 1 $$. The correct answer is:
**(A) both an absolute maximum and an absolute minimum**

---

### Problem 5: Critical Numbers of the Piecewise Function

1. **Find the derivative for each piece:**
   - For $$ x^2 + 1 $$ (if $$ -2 \leq x \leq 1 $$):
     $$
     f'(x) = 2x
     $$
   - For $$ 3x^2 - 4x + 3 $$ (if $$ 1 < x \leq 3 $$):
     $$
     f'(x) = 6x - 4
     $$

2. **Set each derivative to zero:**
   - For $$ 2x = 0 $$:
     $$
     x = 0
     $$
   - For $$ 6x - 4 = 0 $$:
     $$
     x = \frac{2}{3}
     $$

3. **Check the transition point $$ x = 1 $$:**
   The critical numbers are $$ 0 $$ and $$ 1 $$ (the transition point). The correct answer is:
**(A) 0 and 1**

---

### Problem 6: Critical Numbers of $$ f'(x) = x \sin^2 x - \frac{1}{x} $$

1. **Set the derivative to zero:**
   \[
   x \sin
**Problem 1:** Critical numbers of $$ g(x) = \sin x + \cos x $$ on $$ (0, 2\pi) $$ are $$ \frac{\pi}{4} $$ and $$ \frac{5\pi}{4} $$. **Answer:** (C) $$ \frac{\pi}{4} $$ and $$ \frac{5\pi}{4} $$ **Problem 2:** Absolute minimum of $$ f(x) = e^{\sin x} $$ on $$ [0, 2\pi] $$ occurs at $$ \frac{3\pi}{2} $$. **Answer:** (C) $$ \frac{3\pi}{2} $$ **Problem 3:** Maximum value of $$ f(x) = 2x^3 - 15x^2 + 36x $$ on $$ [0, 4] $$ is 32. **Answer:** (C) 32 **Problem 4:** On the interval $$ [0, 1] $$, the function $$ f(x) = 3 - 4x $$ has both an absolute maximum and an absolute minimum. **Answer:** (A) both an absolute maximum and an absolute minimum **Problem 5:** Critical numbers of the piecewise function are 0 and 1. **Answer:** (A) 0 and 1 **Problem 6:** The function $$ f'(x) = x \sin^2 x - \frac{1}{x} $$ has 4 critical numbers on the open interval. **Answer:** (C) 4

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