Question

Give an example of a homogeneous system of three equations in x, y, z that has infinitely many solutions with only one parameter (that is, the solution set is a line). X y || Let f(x) = x³ 3x² - 9x+25. – Use the derivative f'(x) = 3 x² − 6 x − 9 to determine the following values for f on the interval [0, 4].

1. The absolute maximum on this interval is at (x, y (x, y) )

2 The absolute minimum on this interval is at (x, y (x,y) = (₁ = An isosceles triangle with a base of length b and height of h has a rectangle inscribed. If the base of the isosceles triangle is centred on the origin, and the upper right corner of the rectangle is at (x, y), find an expression for the area of the rectangle, A(x), in terms of x, h, and b only. You may find the diagram helpful in solving the problem. You can drag the green dot to see how it affects the area. (x, y) h A(x) = bl X

Answer 1

A homogeneous system of equations with infinitely many solutions representing a line could be one where all equations are scalar multiples of each other. Calculate the function's absolute max and min on an interval by evaluating its derivative at the critical points and the endpoints. The area of an inscribed rectangle in an isosceles triangle is computed using the slope of the sides of the triangle to determine the height of the rectangle as the y-coordinate changes.

Step-by-step explanation:

An example of a homogeneous system of three equations in x, y, z with infinitely many solutions that constitute a line could be:

• x + y + z = 0
• 2x + 2y + 2z = 0
• 3x + 3y + 3z = 0

This system represents three planes in three-dimensional space that are coincident, or overlap each other perfectly, meaning they have all points in common.

To find the absolute maximum and minimum of the function f(x) = x³ + 3x² - 9x + 25 on the interval [0,4], we utilize the derivative f'(x) = 3x² - 6x - 9, find the critical points, and evaluate f(x) at these points as well as the endpoints of the interval.

The formula to calculate the area of the inscribed rectangle in an isosceles triangle with each top side angle having a slope of -h/b and h/b would be A(x) = 2x(h - (h/b)x). This is because the height of the rectangle is the y-value of the upper right corner, which lies on the line with negative slope.