Let's address the questions provided in your uploaded document on differentiation and integration. I’ll go through each question one by one and offer solutions:

Question 1: Curve with equation $y = \sqrt{x} - 4x$

1. Part (a): Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$.

   • To find $\frac{dy}{dx}$, use differentiation rules for both $\sqrt{x}$ (rewriting as $x^{1/2}$) and $-4x$.
   • Then, differentiate $\frac{dy}{dx}$ again to find $\frac{d^2y}{dx^2}$.

2. Part (b): Find the stationary point and determine its nature.

   • Set $\frac{dy}{dx} = 0$ and solve for $x$ to find the stationary points.
   • Use the second derivative test to classify the nature of the stationary points.

Question 2: Square Prism Problem

1. Part (a): Show that $l = \frac{k - 2x^2}{4x}$.

   • Express the length $l$ in terms of $x$ and $k$ by considering the total surface area of the prism.

2. Part (b): Prove that the maximum volume occurs when the prism is a cube.

   • Use the volume formula for the prism, $V = x^2 \cdot l$, substitute for $l$, and differentiate with respect to $x$.
   • Set $\frac{dV}{dx} = 0$ and solve, showing that $l = x$ when volume is maximized.

Question 3: Integrate each expression

1. Part (a): $\int 4x^3 + 3 \, dx$
2. Part (b): $\int x^4 + 5x^2 \, dx$
3. Part (c): $\int (x - 2)^2 \, dx$
4. Part (d): $\int \left(2 + \frac{1}{x^2}\right) dx$

Each integral can be evaluated using basic power and constant integration rules.

Question 8: Evaluate definite integrals

1. Part (a): $\int_1^3 (4x - 1) \, dx$
2. Part (b): $\int_{-2}^{-1} \left(6x + \frac{4}{x^3}\right) \, dx$

Use the fundamental theorem of calculus by finding the antiderivative and then evaluating at the specified limits.

Question 9: Given $\int_1^4 (3x^2 + ax - 5) \, dx = 18$, find $a$.

1. Integrate $3x^2 + ax - 5$ to find an expression in terms of $a$.
2. Evaluate this integral from 1 to 4 and set it equal to 18, then solve for $a$.

Would you like a detailed step-by-step solution for each question or specific ones?

---

Related Questions

1. How would you find the maximum or minimum value of a function using the second derivative test?
2. How is the definite integral of a function between two points evaluated?
3. What are the applications of finding stationary points in real-world scenarios?
4. Can you explain why the volume is maximized when the prism is a cube?
5. How would you solve an integral involving a rational function?

---

Tip:

For complex integrals, breaking down the integrand into simpler parts often makes integration easier.