https://file.aiquickdraw.com/mathbot/file/1729854337918buna9js4.docx

To solve Section C in detail, I'll address each question step by step.

### Question 26(a):
**Problem:** Find \( \frac{dy}{dx} \) if \( y = x \sin x \).

**Solution:**
We need to differentiate \( y = x \sin x \) with respect to \( x \). Using the product rule of differentiation, which states \( \frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v' \), let:
- \( u = x \), so \( u' = 1 \)
- \( v = \sin x \), so \( v' = \cos x \)

Applying the product rule:
\[
\frac{dy}{dx} = \frac{d}{dx}(x \sin x) = (1)(\sin x) + (x)(\cos x) = \sin x + x \cos x
\]

Thus, \( \frac{dy}{dx} = \sin x + x \cos x \).

### Question 27(a):
**Problem:** Find the intervals where the function \( f(x) = x^3 + 6x^2 + 9x - 8 \) is increasing and decreasing.

**Solution:**
1. **Find the derivative** of \( f(x) \):
   \[
   f'(x) = 3x^2 + 12x + 9
   \]

2. **Find the critical points** by setting \( f'(x) = 0 \):
   \[
   3x^2 + 12x + 9 = 0
   \]
   Divide the equation by 3:
   \[
   x^2 + 4x + 3 = 0
   \]
   Factorize the quadratic:
   \[
   (x + 3)(x + 1) = 0
   \]
   Thus, \( x = -3 \) and \( x = -1 \) are the critical points.

3. **Determine the sign of \( f'(x) \)** in the intervals \( (-\infty, -3) \), \( (-3, -1) \), and \( (-1, \infty) \):
   - For \( x < -3 \), choose \( x = -4 \):
     \[
     f'(-4) = 3(-4)^2 + 12(-4) + 9 = 48 - 48 + 9 = 9 > 0
     \]
     So, \( f(x) \) is increasing on \( (-\infty, -3) \).
   - For \( -3 < x < -1 \), choose \( x = -2 \):
     \[
     f'(-2) = 3(-2)^2 + 12(-2) + 9 = 12 - 24 + 9 = -3 < 0
     \]
     So, \( f(x) \) is decreasing on \( (-3, -1) \).
   - For \( x > -1 \), choose \( x = 0 \):
     \[
     f'(0) = 3(0)^2 + 12(0) + 9 = 9 > 0
     \]
     So, \( f(x) \) is increasing on \( (-1, \infty) \).

### Question 28:
**Problem:** Evaluate \( \int \frac{x^2}{x^4 + x^2 - 2} \, dx \).

**Solution:**
1. Factor the denominator:
   \[
   x^4 + x^2 - 2 = (x^2 + 2)(x^2 - 1)
   \]
   Further factor \( x^2 - 1 \) as:
   \[
   (x^2 - 1) = (x - 1)(x + 1)
   \]
   Hence, the integral becomes:
   \[
   \int \frac{x^2}{(x^2 + 2)(x - 1)(x + 1)} \, dx
   \]
2. Use partial fraction decomposition to express \( \frac{x^2}{(x^2 + 2)(x - 1)(x + 1)} \) as a sum of simpler fractions and integrate each term separately.

### Question 29(a):
**Problem:** Find \( \int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} \, dx \).

**Solution:**
1. Let \( I = \int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} \, dx \).
2. Use symmetry or suitable substitutions (e.g., \( x \to \frac{\pi}{2} - x \)) to simplify the integral.
3. Evaluate the resulting expression.

### Question 30:
**Problem:** Find the maximum value of the objective function \( Z = 5x + 10y \) subject to the constraints:
   \[
   x + 2y \leq 120, \quad x + y \geq 60, \quad x - 2y \geq 0, \quad x \geq 0, \quad y \geq 0
   \]

**Solution:**
1. Plot the constraints on a graph to find the feasible region.
2. Determine the vertices of the feasible region.
3. Evaluate \( Z = 5x + 10y \) at each vertex to find the maximum value.

Let me know if you need more details on a particular problem or have further questions.

### Related Questions:
1. How can you determine whether a function is increasing or decreasing?
2. What is the significance of critical points in calculus?
3. How do you factorize higher-order polynomials?
4. Why is partial fraction decomposition useful in integration?
5. What methods can be used to solve linear programming problems graphically?

### Tip:
Always check the endpoints when finding the maximum or minimum of a function on a closed interval.

Get a comprehensive solution for Section C from a Grade 12 Mathematics sample question paper. This includes detailed steps for finding derivatives, analyzing intervals of functions, evaluating integrals, and solving linear programming problems.

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