1. Problem: Show that if $z = f(x + ay) + \Phi(x - ay)$, then $$\frac{\partial^2 z}{\partial y^2} = a^2 \frac{\partial^2 z}{\partial x^2}.$$ 2. Formula and rules: Use the chain rule for partial derivatives. Let $u = x + ay$ and $v = x - ay$. Then $z = f(u) + \Phi(v)$. 3. Compute first derivatives: $$\frac{\partial z}{\partial x} = f'(u) \frac{\partial u}{\partial x} + \Phi'(v) \frac{\partial v}{\partial x} = f'(u) + \Phi'(v)$$ $$\frac{\partial z}{\partial y} = f'(u) \frac{\partial u}{\partial y} + \Phi'(v) \frac{\partial v}{\partial y} = a f'(u) - a \Phi'(v)$$ 4. Compute second derivatives: $$\frac{\partial^2 z}{\partial x^2} = \frac{\partial}{\partial x} (f'(u) + \Phi'(v)) = f''(u) \frac{\partial u}{\partial x} + \Phi''(v) \frac{\partial v}{\partial x} = f''(u) + \Phi''(v)$$ $$\frac{\partial^2 z}{\partial y^2} = \frac{\partial}{\partial y} (a f'(u) - a \Phi'(v)) = a f''(u) \frac{\partial u}{\partial y} - a \Phi''(v) \frac{\partial v}{\partial y} = a^2 f''(u) + a^2 \Phi''(v)$$ 5. Conclusion: Since $$\frac{\partial^2 z}{\partial y^2} = a^2 (f''(u) + \Phi''(v)) = a^2 \frac{\partial^2 z}{\partial x^2},$$ the required relation is proved. 2. Problem: (a) Show that if $u = f(r,s,t)$ where $r = \frac{x}{y}$, $s = \frac{y}{z}$, $t = \frac{z}{x}$, then $$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 0.$$ (b) Verify $$\frac{\partial^2 u}{\partial x \partial y} = \frac{\partial^2 u}{\partial y \partial x}$$ for $u = \tan^{-1} \left(\frac{x}{y}\right)$. 3. For (a), use chain rule and product rule to express derivatives of $u$ in terms of $r,s,t$ and their derivatives. Then simplify to show the sum is zero. For (b), compute first derivatives: $$\frac{\partial u}{\partial x} = \frac{1}{1 + (x/y)^2} \cdot \frac{1}{y} = \frac{y}{x^2 + y^2}$$ $$\frac{\partial u}{\partial y} = \frac{1}{1 + (x/y)^2} \cdot \left(-\frac{x}{y^2}\right) = -\frac{x}{x^2 + y^2}$$ Then compute mixed second derivatives and verify equality. 3. Problem: (a) Discuss maximum and minimum of $$f(x,y) = x^3 y^2 (1 - x - y).$$ (b) Divide 24 into three parts such that the product of the first, square of the second, and cube of the third is maximum. 4. For (a), find critical points by setting partial derivatives to zero: $$\frac{\partial f}{\partial x} = 0, \quad \frac{\partial f}{\partial y} = 0.$$ Use second derivative test to classify maxima/minima. For (b), let parts be $x,y,z$ with $x + y + z = 24$. Maximize $$P = x \cdot y^2 \cdot z^3$$ using Lagrange multipliers or substitution. 4. Problem: (a) Given $$u = \frac{x + y}{1 - xy}, \quad v = \tan^{-1} x + \tan^{-1} y,$$ find $$\frac{\partial(u,v)}{\partial(x,y)}$$ and prove $u,v$ are functionally dependent. Find the relation. (b) Maximize $$x^2 y^2 z^4$$ subject to $$2x + 3y + 4z = a.$$ 5. For (a), compute Jacobian determinant: $$\frac{\partial(u,v)}{\partial(x,y)} = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}.$$ Show it equals zero to prove dependence. Find explicit relation by algebraic manipulation. For (b), use Lagrange multipliers to maximize with constraint. 5. Problem: (a) Show $$\beta(m,n) = 2 \int_0^{\pi/2} \sin^{2m-1} \theta \cos^{2n-1} \theta d\theta.$$ (b) Prove $$\int_0^1 x^m (\log_e \frac{1}{x})^n dx = \frac{n!}{(m+1)^{n+1}},$$ where $n$ is a positive integer. (c) Show $$\int_0^\infty x^n e^{-ax} dx = \frac{1}{a^{n+1}} \Gamma(n+1), \quad a > 0.$$ 6. For (a), recall Beta function definition and use substitution to show equality. For (b), use substitution $t = -\log x$ and properties of Gamma function. For (c), use definition of Gamma function and substitution $t = ax$. Final answers are the proofs and equalities as stated.