[THOUGHT] We need to compute several quantities step by step.

1. For part (1), compute matrix exponential e^A for given A. We can compute eigenvalues and use diagonalization or use sympy's expm. Since A is 3x3, we can compute eigenvalues λ1, λ2, λ3, then compute product λ1λ2λ3 and sum of squares λ1^2+λ2^2+λ3^2.

2. For part (2), we have A_x = [[0, -x, 0], [x, 0, 0], [0, 0, 1]] and B = diag(1, -1, 1). Compute C = B * exp(A_x). Then we need min and max of Rayleigh quotient y^T C y / (y^T y) over nonzero y. That's just the minimum and maximum eigenvalues of the symmetric part? But C is not symmetric? Wait: y^T C y / (y^T y) for real y and real C. Since C may not be symmetric, the Rayleigh quotient for non-symmetric matrices is not simply eigenvalues. However, the problem likely expects that C is symmetric? Let's check: A_x is skew-symmetric in the upper left, plus a 1 in bottom right. Its exponential will be rotation in first two coordinates and e^1 in third? Actually exp(A_x) = exp([[0,-x,0],[x,0,0],[0,0,1]]). Since the block is block diagonal: the upper left 2x2 is skew-symmetric, its exponential is rotation matrix: [[cos x, -sin x, 0], [sin x, cos x, 0], [0, 0, e^1]]? Wait, the third diagonal entry is 1, so exp(1) = e. So exp(A_x) = [[cos x, -sin x, 0], [sin x, cos x, 0], [0, 0, e]]. Then multiply by B = diag(1, -1, 1). So C = B exp(A_x) = [[cos x, -sin x, 0], [-sin x, -cos x, 0], [0, 0, e]]? Let's compute: B * exp(A_x) = diag(1,-1,1) * [[cos x, -sin x, 0], [sin x, cos x, 0], [0, 0, e]] = [[cos x, -sin x, 0], [-sin x, -cos x, 0], [0, 0, e]]. Indeed, multiply second row by -1. So C = [[cos x, -sin x, 0], [-sin x, -cos x, 0], [0, 0, e]].

Now C is symmetric! So the Rayleigh quotient's min and max are the smallest and largest eigenvalues of C. So we need eigenvalues of C as functions of x. Then we need min and max over all real x? Wait: The problem says "where x is a real number" and define C = B e^{A_x}. Then "Determine the minimum and maximum values of (y^T C y)/(y^T y) over all nonzero real vectors y". That suggests for each fixed x, the Rayleigh quotient ranges between min and max eigenvalues of C. But the min and max values over all nonzero y for a given C are the eigenvalues of C. However, the problem likely asks: For each fixed x, the min and max of the Rayleigh quotient are the eigenvalues of C. But then they ask "minimum and maximum values" over all nonzero y? That is ambiguous: maybe they mean for a given x, the min and max of the Rayleigh quotient are the eigenvalues of C. But then they ask "Let m and M be the minimum and maximum values, respectively, obtained in part (2)". That suggests m and M are numbers, not depending on x. Possibly they mean: Consider the set of all possible Rayleigh quotients as y varies over nonzero vectors, and x is fixed? Or maybe x is also variable? The wording: "where x is a real number" and define C = B e^{A_x}. Then "Determine the minimum and maximum values of ... over all nonzero real vectors y". That seems to indicate for each fixed x, the Rayleigh quotient's min and max are eigenvalues of C. But then m and M are just those eigenvalues. However, the problem likely expects m and M to be specific numbers, not depending on x. Possibly x is given? Wait, part (2) does not specify a value for x. Maybe x is arbitrary real, and we need the min and max of the Rayleigh quotient over all y and all x? That would be weird because then we would optimize over x as well. Alternatively, maybe the problem expects that we consider the matrix C as defined with x being a parameter, and we need the min and max eigenvalues of C as functions of x, and then perhaps the overall min and max over x? But the phrasing "Determine the minimum and maximum values of ... over all nonzero real vectors y" suggests for a given C, the min and max are eigenvalues. But C depends on x. So maybe they want the min and max eigenvalues of C for any x? That seems plausible: they want the extreme possible values of the Rayleigh quotient as y varies and x varies? Actually the problem says "Let C = B e^{A_x}. Determine the minimum and maximum values of (y^T C y)/(y^T y) over all nonzero real vectors y." That is ambiguous: does "over all nonzero real vectors y" mean for each fixed C, we consider all y, and we get min and max eigenvalues. But then m and M would depend on x. However, the final expression uses m and M as numbers. So likely x is fixed? Wait, maybe x is not specified because it cancels? Let's compute eigenvalues of C symbolically in x.

C = [[cos x, -sin x, 0], [-sin x, -cos x, 0], [0, 0, e]].

The characteristic polynomial: det(C - λI) = det([[cos x - λ, -sin x, 0], [-sin x, -cos x - λ, 0], [0, 0, e - λ]]). This is block diagonal: the upper 2x2 block and the lower 1x1. So eigenvalues are e (since e is constant ≈ 2.718...), and the eigenvalues of the 2x2 block: [[cos x - λ, -sin x], [-sin x, -cos x - λ]]. Compute its determinant: (cos x - λ)(-cos x - λ) - (-sin x)(-sin x) = -(cos x - λ)(cos x + λ) - sin^2 x = -(cos^2 x - λ^2) - sin^2 x = -cos^2 x + λ^2 - sin^2 x = λ^2 - (cos^2 x + sin^2 x) = λ^2 - 1. So eigenvalues of the 2x2 block satisfy λ^2 - 1 = 0 => λ = ±1. So the eigenvalues of C are 1, -1, and e. That is independent of x! So indeed, for any x, the eigenvalues are 1, -1, e. Then the Rayleigh quotient's min and max over nonzero y are the min and max eigenvalues of C (since C is symmetric). So min = -1, max = e (since e > 1). But careful: e is about 2.718, so max is e. But is e exactly the constant e? Yes, it's the base of natural exponential. So m = -1, M = e.

3. For part (3), f(z) = (1/2) z^T C z - a^T z, with a = [1, 1, e]^T. The gradient condition ∂f/∂z = 0 gives C z = a. So z = C^{-1} a. Then we need R = sqrt(z1^2+z2^2+z3^2) = norm of z. Since C has eigenvalues 1, -1, e, it's invertible. Compute z = C^{-1} a. But C depends on x? However, the eigenvalues are independent of x, but the eigenvectors depend on x, so z might depend on x. But the problem likely expects a unique R? Possibly because a is special, maybe R is independent of x. Let's compute symbolically.

C = [[cos x, -sin x, 0], [-sin x, -cos x, 0], [0, 0, e]]. Compute C^{-1}. Since C is block diagonal: inverse of 2x2 block: [[cos x, -sin x], [-sin x, -cos x]] inverse: determinant = -cos^2 x - sin^2 x = -1. So inverse = (1/-1) * [[-cos x, sin x], [sin x, cos x]] = [[cos x, -sin x], [-sin x, -cos x]]? Wait, compute properly: For matrix [[a, b], [c, d]], inverse = 1/(ad-bc) * [[d, -b], [-c, a]]. Here a = cos x, b = -sin x, c = -sin x, d = -cos x. Determinant = ad - bc = cos x * (-cos x) - (-sin x)*(-sin x) = -cos^2 x - sin^2 x = -1. So inverse = -1 * [[-cos x, sin x], [sin x, cos x]] = [[cos x, -sin x], [-sin x, -cos x]]. That is exactly the same as the original 2x2 block! So the 2x2 block is its own inverse? Check: multiply [[cos x, -sin x], [-sin x, -cos x]] by itself: yields [[cos^2 x + sin^2 x, -cos x sin x + sin x cos x], ...] = [[1, 0], [0, cos^2 x + sin^2 x]] = identity? Let's verify with Python later. Actually compute: (cos x)(cos x) + (-sin x)(-sin x) = cos^2 x + sin^2 x = 1. Off-diagonal: (cos x)(-sin x) + (-sin x)(-cos x) = -cos x sin x + sin x cos x = 0. So indeed the 2x2 block is orthogonal? Actually it's symmetric and its square is identity. So indeed C_{2x2} is its own inverse. So C^{-1} = C? Wait, the whole C: the 3,3 entry is e, its inverse is 1/e. So C^{-1} = [[cos x, -sin x, 0], [-sin x, -cos x, 0], [0, 0, 1/e]]. So z = C^{-1} a = [[cos x, -sin x, 0], [-sin x, -cos x, 0], [0, 0, 1/e]] * [1, 1, e]^T = [cos x - sin x, -sin x - cos x, 1]^T. Then R = sqrt((cos x - sin x)^2 + (-sin x - cos x)^2 + 1^2). Compute: (cos x - sin x)^2 = cos^2 x - 2 cos x sin x + sin^2 x = 1 - 2 cos x sin x. (-sin x - cos x)^2 = sin^2 x + 2 sin x cos x + cos^2 x = 1 + 2 sin x cos x. Sum = (1 - 2 sin x cos x) + (1 + 2 sin x cos x) = 2. Then plus 1 gives 3. So R = sqrt(3). Indeed independent of x. So R = sqrt(3).

Now for part (1), compute eigenvalues of A. A = [[-1,4,4],[-5,8,10],[3,-3,-5]]. Compute eigenvalues λ1, λ2, λ3. Then compute product λ1 λ2 λ3 = det(A). And compute sum of squares λ1^2+λ2^2+λ3^2 = (trace(A))^2 - 2 sum of principal minors? Or compute directly.

Then final expression: λ1 λ2 λ3 * m * R^2 * (λ1^2+λ2^2+λ3^2). m = -1, R^2 = 3. So product = det(A) * (-1) * 3 * (λ1^2+λ2^2+λ3^2). So we need det(A), and sum of squares of eigenvalues.

We'll compute using Python.

Let's implement step by step. [/THOUGHT] [PYTHON] import sympy as sp import numpy as np import math

Part (1): Compute eigenvalues of A

A = sp.Matrix([[-1, 4, 4], [-5, 8, 10], [3, -3, -5]]) print("Matrix A:") print(A) print()

Compute eigenvalues

eigvals = A.eigenvals() print("Eigenvalues (with multiplicities):", eigvals)

Get list of eigenvalues

eigvals_list = list(eigvals.keys()) print("Eigenvalues:", eigvals_list)

Compute product (determinant)

det_A = A.det() print("det(A) = product of eigenvalues =", det_A)

Compute sum of squares of eigenvalues

sum_sq = sum(lam**2 for lam in eigvals_list) print("Sum of squares of eigenvalues:", sum_sq)

Part (2): m and M

m = -1, M = e

m = -1 M = sp.E # symbolic e print("\nPart (2): m =", m, ", M =", M)

Part (3): R = sqrt(3)

R = sp.sqrt(3) print("R =", R) R_sq = R**2 print("R^2 =", R_sq)

Compute final expression: λ1 λ2 λ3 * m * R^2 * (λ1^2+λ2^2+λ3^2)

final_expr = det_A * m * R_sq * sum_sq print("\nFinal expression value (symbolic):", final_expr) print("Final expression value (numeric):", final_expr.evalf()) [/PYTHON]

[THOUGHT] The Python output gives eigenvalues 3, 1, -2, so product λ1λ2λ3 = -6, sum of squares = 9+1+4=14. m = -1, R^2 = 3. Then final expression = (-6) * (-1) * 3 * 14 = 6 * 3 * 14 = 18 * 14 = 252. The symbolic and numeric output confirm 252. [/THOUGHT] \boxed{252}