He2 has a bond order of 0, as it has equal bonding and antibonding electrons.

The molecular orbital diagram of F2 is similar to that of O2, with the same energy level arrangement.

O2 and F2 both have a bond order of 2.

To ensure differentiability at x = -1, we find f'(-1) exists. Setting a = 0 ensures the derivative is defined.

The function is a polynomial and is differentiable for all real numbers, hence any value of a works.

Setting the derivative f'(1) = 0 gives a = 1 for differentiability.

f(x) is a polynomial and is differentiable for all a, hence any value of a works.

Setting f'(1) = 0 gives a = 1, ensuring differentiability at that point.

Setting f'(1) = 0 gives a = 1 for differentiability at x = 1.

Setting a = 1 gives continuity at x = 0.

Setting 3(2) + a = 4(2) - 1 gives a = 1.

Setting ax + 1 = 2 at x = 0 gives a = 2.

Setting the two pieces equal at x = 0 gives 1 = a, hence a = 1.

Setting ax + 2 = 3 at x = 1 gives a = 1.

To ensure continuity at x = 1, we set limit as x approaches 1 from left (1 + a) equal to f(1) = 3, thus a = 2.

Setting the two pieces equal at x = 0 gives -a = 1, so a = -1.

Setting the two pieces equal at x = 1 gives 1 - a = 1. Thus, a = 0.

Setting the two pieces equal at x = 1 gives 1 = a(1) + 1, leading to a = 0.

Setting the left limit (2(1) + 1 = 3) equal to the right limit (1^2 + 1 = 2), we find b = 3.

Setting limit as x approaches 1 gives b = 2 for continuity.

Setting the two pieces equal at x = 2 gives us 0 = 2b + 2. Solving for b gives b = -1.

Setting 1 - 3 + b = 2 gives b = 4 for continuity.

Setting 3(1) + c = 2(1)^2 gives c = -1 for continuity.

Setting the two pieces equal at x = c: c^2 - 4 = 3c - 5. Solving gives c = 3.

Setting x^2 - c = 2x + 1 at x = 1 gives c = 2.

Setting x^2 = 2x + 1 at x = c gives c = 2.

The discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => k < 8 and k > -8.

The discriminant must be positive: k^2 - 4*1*16 > 0 => k^2 > 64 => k > 8 or k < -8. Thus, k = -4 is valid.

Substituting x = 2 into the equation gives 2^2 + 2k + 4 = 0, leading to k = -4.

Let the roots be r and r+2. Then, r + (r+2) = 4 and r(r+2) = k leads to k = 4.