Calculus Optimization Challenge

What is the maximum value of the function f(x) = -2x^2 + 4x + 1?

• 4

• 1

• 2

• 3

Find the minimum point of the function g(x) = x^3 - 3x^2 + 4.

• (0, 4)

• (2, 0)

• (-1, 6)

• (√2, 2 - 2√2)

Determine the critical points of the function h(x) = 3x^4 - 8x^3 + 6.

• x = 1

• x = -2

• x = 0, x = 2

• x = 3

Using calculus, find the dimensions of a rectangle with a perimeter of 20 that maximizes the area.

• 2, 8

• 5, 5

• 3, 7

• 4, 6

A farmer wants to fence a rectangular area using 100 meters of fencing. What dimensions will maximize the area?

• 25 meters by 25 meters

• 20 meters by 30 meters

• 15 meters by 35 meters

• 10 meters by 40 meters

Find the point on the curve y = x^2 + 2x that is closest to the point (1,0).

• (0, 0)

• (1, 1)

• (2, 4)

• (-1, 1)

What is the maximum profit if the profit function is given by P(x) = -5x^2 + 50x - 100?

• 30

• 20

• 15

• 25

Using the first derivative test, determine the local maxima and minima of the function j(x) = x^4 - 8x^2 + 16.

• Local maximum at x = 2, local minimum at x = 0.

• Local maximum at x = 0, local minimum at x = -2.

• Local maximum at x = -2, local minimum at x = 2.

• Local maximum at x = 1, local minimum at x = -1.

Find the dimensions of a box with a square base that has a volume of 500 cubic meters and minimizes the surface area.

• Base: 5 meters, Height: 20 meters

• Base: 10 meters, Height: 5 meters

• Base: 15 meters, Height: 2.22 meters

• Base: 8 meters, Height: 12.5 meters

If the revenue function is R(x) = 100x - 0.5x^2, find the number of units sold that maximizes revenue.

• 100

• 150

• 50

• 200