The problem asks to determine whether each function has a maximum, minimum, or both. It also asks to evaluate the critical points of the functions where these extrema occur (where the derivative equals zero) and to determine points of inflection.

I'll go through the process of solving each of these functions:

a) $y = 4x^2 - 5x + 4$

This is a quadratic function. To find the critical points, take the derivative and set it equal to zero: $\frac{dy}{dx} = 8x - 5$ Setting $8x - 5 = 0$, we get $x = \frac{5}{8}$.

Now, check if this is a maximum or minimum by using the second derivative: $\frac{d^2y}{dx^2} = 8$ Since $\frac{d^2y}{dx^2} > 0$, this critical point is a minimum.

b) $y = 2x^3 + 3x^2 - 36x + 16$

Take the derivative: $\frac{dy}{dx} = 6x^2 + 6x - 36$ Set the derivative equal to zero: $6(x^2 + x - 6) = 0 \quad \Rightarrow \quad x^2 + x - 6 = 0$ Factoring gives: $(x - 2)(x + 3) = 0$ So, $x = 2$ and $x = -3$.

Now, check the second derivative: $\frac{d^2y}{dx^2} = 12x + 6$ For $x = 2$, $\frac{d^2y}{dx^2} = 12(2) + 6 = 30 > 0$, so $x = 2$ is a minimum.

For $x = -3$, $\frac{d^2y}{dx^2} = 12(-3) + 6 = -30 < 0$, so $x = -3$ is a maximum.

c) $y = \sin(3x)$

Take the derivative: $\frac{dy}{dx} = 3\cos(3x)$ Setting $3\cos(3x) = 0$, gives $\cos(3x) = 0$, so: $3x = \frac{\pi}{2} + n\pi \quad \Rightarrow \quad x = \frac{\pi}{6} + \frac{n\pi}{3}$ The points where maxima and minima occur are periodic.

d) $\psi = Ae^{-mx}$

This is an exponential decay function. Since it's monotonically decreasing (assuming $m > 0$), it has no maximum or minimum.

e) $U(r) = e \left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right]$

To find critical points, you would need to take the derivative with respect to $r$. This function describes the Lennard-Jones potential, typically having a minimum at a specific point, representing the equilibrium distance between particles.

f) $\psi = \frac{1}{2} (1 + \sin(\theta)) + \sqrt{2}\cos(\theta)$

Differentiate with respect to $\theta$: $\frac{d\psi}{d\theta} = \frac{1}{2} \cos(\theta) - \sqrt{2} \sin(\theta)$ Set the derivative equal to zero and solve for $\theta$ to find critical points.

g) $E = \frac{e^2}{a}\left(\frac{z - 27}{8z}\right)$

This requires finding the derivative of $E$ with respect to $z$, then setting it equal to zero to find critical points.

h) $P_E = \frac{2\sqrt{E}}{kT\sqrt{\pi kT}} e^{-E/kT}$

Take the derivative with respect to $E$ to find critical points.

i) $P(x) = \left(\frac{2}{a}\right)\sin^2\left(\frac{\pi x}{a}\right)$

Take the derivative of $P(x)$ with respect to $x$, then set it equal to zero to find critical points.

j) $U(r) = -N_0 A\frac{z^2}{r} + \frac{B}{r^n}$

Take the derivative of $U(r)$ with respect to $r$, and set it equal to zero to find critical points.

Would you like further details on any specific part or function? Here are some follow-up questions:

1. How do we determine if a function has a point of inflection?
2. What methods can be used to classify critical points beyond second derivatives?
3. How do periodic functions behave when finding maxima and minima?
4. What is the physical interpretation of critical points in the Lennard-Jones potential?
5. How can numerical methods help in finding critical points for complex functions?

Tip: Always use the second derivative test to classify critical points efficiently!