Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

This positive number is 1/2. First we construct an equation that represents what is being described, with x being our number to find: y = 1/x + 4*x^2 Take the derivative: y’ = -1/(x^2) + 8x Our number would be the value of x where the derivative of y = 0: 0 = -1/(x^2) + 8x Solve for x: 1/(x^2) = 8x 1 = 8x^3 1/8 = x^3 x = 1/2