The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 7 = 3.

f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(12x - 24) = 0, so x = 0 or x = 2. Check f(1) = 3.

Set f'(x) = 0. f'(x) = 6x^2 - 18x + 12 = 0 gives x = 2. f(2) = 0.

The maximum occurs at x = 2, found by setting f'(x) = 4 - 2x = 0. f(2) = 4(2) - (2^2) = 4.

The product of the roots is given by c/a = 8/1 = 8.

The product of the roots is c/a = 9/1 = 9.

Factoring gives (x + 4)(x - 2) = 0, hence the roots are -4 and 2.

This is a perfect square: (x + 3)² = 0, hence the root is x = -3.

During inhalation, the diaphragm contracts, increasing the volume of the thoracic cavity and decreasing the pressure, allowing air to flow in.

During metaphase, chromosomes align at the metaphase plate, preparing for separation.

Oxygen enters the bloodstream during inhalation when air is drawn into the lungs and diffuses across the alveolar membrane.

During exhalation, the diaphragm relaxes, causing the thoracic cavity to decrease in volume and air to be expelled from the lungs.

Ovulation occurs during the ovulatory phase of the menstrual cycle when an egg is released from the ovary.

The endometrium thickens during the luteal phase of the menstrual cycle in preparation for a potential pregnancy.

The embryo implants into the uterine wall during the blastocyst stage of development.

Ethylene is produced in response to low oxygen levels, particularly during stress conditions like flooding.

The integral is (3/3)x^3 + (2/2)x^2 + C = x^3 + x^2 + C.

Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.

Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.

Using the Taylor series expansion for sin(x), we find that lim (x -> 0) (x - sin(x))/x^3 = 1/6.

As x approaches 0, x^3 approaches 0 and sin(x) approaches 0, thus the limit is 0.

Factoring gives (x - 3)(x + 3)/(x - 3). Canceling (x - 3) gives lim (x -> 3) (x + 3) = 6.

The integral is (2/2)x^2 + 3x + C = x^2 + 3x + C.

The integral is (4/4)x^4 - (2/2)x^2 + C = x^4 - x^2 + C.

The integral is 5x - (3/2)x^2 + C.

Integrating term by term: ∫2x^2dx = (2/3)x^3, ∫3xdx = (3/2)x^2, and ∫1dx = x. Thus, ∫(2x^2 + 3x + 1)dx = (2/3)x^3 + (3/2)x^2 + x + C.

The integral of 5x^4 is (5/5)x^5 + C = x^5 + C.

Integrating term by term: ∫6x^2dx = 2x^3 and ∫3dx = 3x. Thus, ∫(6x^2 + 3)dx = 2x^3 + 3x + C.

Faraday's law states that the induced EMF is proportional to the rate of change of magnetic flux.

Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.