The length of the latus rectum for the parabola y^2 = 4px is given by 4p. Here, p = 4, so the length is 16.

f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = ±1. f''(1) = 6 > 0 (min), f''(-1) = 6 > 0 (min). Local maxima at (0, 0).

f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1, 3. f''(1) > 0 (min), f''(3) < 0 (max).

f'(x) = 4x^3 - 12x^2 + 4. Setting f'(x) = 0 gives x = 0 and x = 2. f''(0) = 4 > 0 (min), f''(2) = -8 < 0 (max).

The vertex occurs at x = 2. f(2) = -2^2 + 4(2) + 1 = 5, which is the maximum value.

The vertex occurs at x = 2. f(2) = 2^2 - 4*2 + 5 = 1. Thus, the minimum value is 1.

The moment of inertia of a solid sphere about an axis through its center is I = 2/5 MR^2.

The discriminant indicates that the lines intersect at two distinct points.

Find f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(x - 2) = 0, so x = 0 or x = 2. f''(2) = 6 > 0, so (2, 1) is a local minimum.

The function |x - 1| is not differentiable at x = 1 due to a cusp.

The function f(x) = |x - 3| is not differentiable at x = 3 because it has a sharp corner.

f(x) is not differentiable at x = -2 and x = 2, but is differentiable everywhere else.

f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x = 0 and x = 2. The point of inflection is at (1, 3).

Find f''(x) = 12x^2 - 24x + 12. Setting f''(x) = 0 gives x = 1 and x = 2. Testing intervals shows a change in concavity at x = 1.

Setting 2x + 1 = -x + 4 gives 3x = 3, hence x = 1. Substituting back gives y = 3.

The function is a polynomial and is differentiable everywhere, hence nowhere.

The function is a polynomial and is differentiable everywhere. Thus, there are no points where it is not differentiable.

Scalar product = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.

A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.

A · B = 2*3 + 2*3 + 2*3 = 6 + 6 + 6 = 18.

-2x + 6 > 0 => -2x > -6 => x < 3.

-2x + 6 ≥ 0 => -2x ≥ -6 => x ≤ 3.

-3x + 1 ≤ 4 => -3x ≤ 3 => x ≥ -1.

-3x + 4 ≤ 1 => -3x ≤ -3 => x ≥ 1.

2x + 3 ≤ 7 => 2x ≤ 4 => x ≤ 2.

6 - x > 2 => -x > -4 => x < 4.

6 - x ≤ 3 => -x ≤ -3 => x ≥ 3.

7 - 3x < 1 => -3x < -6 => x > 2.

7x - 2 ≤ 5x + 6 => 2x ≤ 8 => x ≤ 4.

2(x - 1) ≥ 3 => x - 1 ≥ 1.5 => x ≥ 2.