Homework

Problem 1 · easy

Find the rectangle of maximum area with perimeter 20.

Problem 2 · easy

Find the number whose square is maximized given x + y = 10 and you want to maximize xy.

Problem 3 · easy

Two non-negative numbers sum to 12. Find the numbers that maximize their product.

Problem 4 · medium

Find the minimum value of f(x) = x + 4/x for x > 0.

Problem 5 · medium

Find the minimum perimeter of a rectangle with area 36.

Problem 6 · medium

An open-top box with square base holds 108 cm^3. Find the base side that minimizes surface area.

Problem 7 · challenge

A farmer encloses a rectangular area of 200 m^2 using a wall on one side. Find the minimum fence length.

Problem 8 · medium

Find the point on y = x closest to the point (3, 0). Give the x-coordinate.

Problem 9 · challenge

Find the point on y = x^2 + 1 closest to the origin. Give the x-coordinate.

Problem 10 · medium

A company has revenue R = 100x - 2x^2 and cost C = 20x + 300. What x maximizes profit?

Problem 11 · medium

Find the maximum profit from Problem 10.

Problem 12 · challenge

The sum of two positive numbers is 8. Maximize the product of one with the square of the other.

Problem 13 · medium

Find the minimum value of 2x + 18/x for x > 0.

Problem 14 · easy

A projectile is launched at 45 degrees. The range formula is R = v^2*sin(2*theta)/g. What angle gives max range?

Problem 15 · challenge

An isosceles triangle has base 8 and two equal sides of length s. Maximize the area in terms of s (set up dA/ds = 0).

Problem 16 · easy

Find the x that minimizes f(x) = (x-2)^2 + 1.

Problem 17 · medium

Find the absolute max of f(x) = x^3 - 3x on [-2, 2].

Problem 18 · challenge

Minimize S = 4x + 9/x^2 for x > 0.

Problem 19 · challenge

A rectangle is inscribed in a circle of radius 5. Find the maximum area.

Problem 20 · medium

Two positive numbers multiply to 16. Minimize their sum.

Problem 21 · challenge

Find the cheapest box with square base, open top, volume 256 cm^3 if base costs $2/cm^2 and sides cost $1/cm^2.

Problem 22 · easy

A fence 100 m long encloses a rectangle. Find the length that maximizes area.

Problem 23 · easy

What is the maximum area from Problem 22?

Problem 24 · medium

Find the minimum of f(x) = x^4 - 8x^2 + 5.

Problem 25 · challenge

A window has a rectangle topped by a semicircle. Perimeter is 12 m. Maximize the area. (Express max area in terms of pi.)