f'(1) from left = 2, from right = 2; hence f is differentiable at x = 1.

By substituting x = 1 into the equation, we find that it satisfies the equation, hence 1 is a root.

The roots can be found using the Rational Root Theorem and synthetic division, confirming that all roots are real.

Mean = 30; Mean Deviation = (|10-30| + |20-30| + |30-30| + |40-30| + |50-30|) / 5 = 10.

The mode is 4, as it appears 3 times, more than any other number.

The mode is the number that appears most frequently, which is 23.

There is no mode as all values appear only once.

Mean = 6; Mean deviation = (|2-6| + |4-6| + |6-6| + |8-6| + |10-6|)/5 = (4 + 2 + 0 + 2 + 4)/5 = 12/5 = 2.4.

Mean = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.0.

Q1 = 2, Q3 = 4; Interquartile Range = Q3 - Q1 = 4 - 2 = 2.

Mean = (1 + 2 + 3 + 4 + 5) / 5 = 3. Variance = [(1-3)² + (2-3)² + (3-3)² + (4-3)² + (5-3)²] / 5 = 2.

Mean = (10 + 12 + 14 + 16 + 18) / 5 = 14. Variance = [(10-14)² + (12-14)² + (14-14)² + (16-14)² + (18-14)²] / 5 = 8.

Mean = (10 + 12 + 14 + 16) / 4 = 13. Variance = [(10-13)² + (12-13)² + (14-13)² + (16-13)²] / 4 = 2.

Mean = (2 + 4 + 6 + 8 + 10) / 5 = 6. Variance = [(2-6)² + (4-6)² + (6-6)² + (8-6)² + (10-6)²] / 5 = 8.

All values are the same, so variance = 0.

Mean = 7.5; MAD = (|5-7.5| + |7-7.5| + |8-7.5| + |9-7.5| + |10-7.5|) / 5 = 1.

Mean = 7.5; Variance = [(5-7.5)^2 + (7-7.5)^2 + (8-7.5)^2 + (9-7.5)^2 + (10-7.5)^2] / 5 = 2; Standard Deviation = sqrt(2) = 1.41

Mean = 9. Variance = [(5-9)² + (7-9)² + (9-9)² + (11-9)² + (13-9)²] / 5 = 8.

Arranging the numbers: 5, 8, 12, 15, 20. The median is the middle number, which is 12.

The mode is 9, as it appears most frequently (three times) in the data set.

The semi-major axis is 4 and the semi-minor axis is 3 after rewriting the equation in standard form.

The discriminant is 2^2 - 4(1)(1) = 0, indicating that the roots are real and equal.

Substituting x = 1 into the equation gives 1^2 + 2*1 + k = 0 => 1 + 2 + k = 0 => k = -3.

The discriminant must be non-negative: 4^2 - 4*1*k >= 0 leads to 16 - 4k >= 0, thus k <= 4.

The condition for no real roots is that the discriminant must be less than zero: 6^2 - 4*1*k < 0 => 36 < 4k => k > 9.

The discriminant must be non-negative: k^2 - 4*1*9 >= 0 => k^2 >= 36 => k <= -6 or k >= 6.

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

The equation can be factored as (x-1)^3 = 0, which has one real root (x = 1) with multiplicity 3.

By substituting x = 2 into the equation, we find that it satisfies the equation, thus x = 2 is a root.

The product of the roots of the cubic equation ax^3 + bx^2 + cx + d = 0 is given by -d/a. Here, d = -6 and a = 1, so the product is -(-6)/1 = 6.