Question

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

Answer 1

To find the positive number that minimizes the sum of its reciprocal and four times its square, we use calculus to find the derivative and solve for the critical points. However, there is no positive number that satisfies the given condition.

Step-by-step explanation:

To find a positive number for which the sum of its reciprocal and four times its square is the smallest possible, we can solve this problem using calculus. Let's denote the positive number as x. We can set up the equation as follows:

x + 4x^2 = y

To minimize y, we need to find the value of x that makes the derivative of y with respect to x equal to 0. Differentiating y with respect to x, using the power rule, we get:

1 + 8x = 0

Solving this equation for x, we find x = -1/8.

Since we are looking for a positive number, x = -1/8 is not a valid solution. Therefore, there is no positive number for which the sum of its reciprocal and four times its square is the smallest possible.

Answer 2

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