At x = 2, left limit is 4 and right limit is 4, but f(2) = 4. Hence, f(x) is continuous at x = 2.

The left limit as x approaches 3 is 9, the right limit is also 9, and f(3) = 9. Therefore, f(x) is continuous at x = 3.

The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10. The adjugate is [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].

Calculating J^2 gives [[0, 1], [1, 0]] * [[0, 1], [1, 0]] = [[1, 0], [0, 1]], which is the identity matrix.

Using the determinant formula, det(J) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) + 3*(-5) = -24 + 40 - 15 = 1.

Det(F) = (2*4) - (1*3) = 8 - 3 = 5.

The equation x^2 = -12y indicates that the parabola opens downwards.

In the equation x^2 = 4py, we have 4p = 16, thus p = 4. The distance from the vertex to the focus is 4.

The length of the latus rectum for the parabola x^2 = 4py is 4p. Here, p = 4, so the length is 8.

To find the y-intercept, set x = 0. The equation becomes y = -0^2 + 4(0) - 3 = -3.

The polynomial can be factored as (x-1)^3, indicating that it has one real root with multiplicity 3, hence all roots are real and equal.

The sum of the roots is given by -b/a = 3/1 = 3.

The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.

For equal roots, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(2)(k) = 0 leads to k = 4.

For real and equal roots, the discriminant must be zero. Here, b^2 - 4ac = 0 gives 16 - 8k = 0, thus k = 8.

The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(-6) = 16 + 48 = 64.

For equal roots, the discriminant must be zero: (-4)^2 - 4*2*k = 0. Solving gives k = 4.

Using the quadratic formula x = [-b ± √(b^2 - 4ac)] / 2a, we find the roots to be -1 and 2/5.

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.

The discriminant must be non-negative: 2^2 - 4*1*k >= 0 leads to k <= 1.

The discriminant D = 6^2 - 4*1*9 = 0, indicating real and equal roots.

The discriminant must be positive: 6^2 - 4*1*k > 0, which simplifies to k < 9.

The discriminant must be non-negative: 6^2 - 4(1)(k) ≥ 0, which gives k ≤ 9.

The sum of the roots is -(-2) + -(-3) = 5, hence p = 5.

The discriminant is 0, indicating that the roots are real and equal.

If one root is 3, then substituting x = 3 gives 3^2 - 6*3 + k = 0, leading to k = 9.

The roots can be found by factorization: (x - 3)(x - 5) = 0, hence the roots are 3 and 5.

A · B = (2)(5) + (3)(6) = 10 + 18 = 28.