Final answer:
To find the positive number that minimizes the expression, set up an equation, take the derivative, and solve for x. The positive number that minimizes the expression is approximately 0.583.
Step-by-step explanation:
To find the positive number for which the sum of its reciprocal and ten times its square is the smallest possible, we need to set up an equation and find the number that minimizes that equation. Let's denote the positive number as x.
The sum of its reciprocal and ten times its square can be written as 1/x + 10x^2. To find the minimum value, we need to find the value of x that makes the derivative of this expression equal to zero. Taking the derivative and setting it equal to zero, we get 1/-x^2 + 20x = 0. Multiplying through by -x^2, we get -1 + 20x^3 = 0.
Simplifying the equation, we have 20x^3 = 1. Dividing both sides by 20, we find that x^3 = 1/20. Taking the cube root of both sides, we get x = (1/20)^(1/3), which is approximately 0.583. Therefore, the positive number that minimizes the expression is 0.583.