Find a positive number such that the sum of the square of the number and its reciprocal is a minimum.
let the number be x, and the sum as stated be S
S = x^2 + 1/x
dS/dx = 2x - 1/x^2
= 0 for a max/min
2x = 1/x^2
2x^3 = 1
x^3 = 1/2
x = (1/2)^(1/3) or the cube root of 1/2
Why did the number go to therapy?
Because it couldn't find its reciprocal and it felt so square!
To find the positive number that minimizes the sum of its square and reciprocal, we can use calculus. Let's call the positive number x.
Step 1: Write the equation for the sum of the square of the number and its reciprocal:
f(x) = x^2 + 1/x
Step 2: Take the derivative of f(x) with respect to x:
f'(x) = 2x - 1/x^2
Step 3: Set f'(x) equal to 0 and solve for x:
2x - 1/x^2 = 0
Step 4: Multiply through by x^2 to get rid of the fraction:
2x^3 - 1 = 0
Step 5: Add 1 to both sides:
2x^3 = 1
Step 6: Divide both sides by 2:
x^3 = 1/2
Step 7: Take the cube root of both sides:
x = (1/2)^(1/3) or x ≈ 0.7937
So, the positive number that minimizes the sum of its square and reciprocal is approximately 0.7937.
To find the positive number that minimizes the sum of its square and reciprocal, we can use calculus. Let's call the positive number "x".
Step 1: Set up the equation to minimize
We want to minimize the expression x^2 + 1/x. So, the equation we need to minimize is f(x) = x^2 + 1/x.
Step 2: Find the derivative of the equation
To find the minimum, we need to find the critical points by setting the derivative of f(x) equal to zero. So, let's find f'(x):
f'(x) = 2x - 1/x^2
Step 3: Set f'(x) equal to zero and solve
Setting f'(x) = 0 gives us:
2x - 1/x^2 = 0
Multiplying through by x^2:
2x^3 - 1 = 0
Rearranging the equation:
2x^3 = 1
x^3 = 1/2
Taking the cube root of both sides:
x = (1/2)^(1/3)
x ≈ 0.7937
Step 4: Determine if the critical point is a minimum
To confirm that x = 0.7937 is the number that minimizes the expression, we can check if it is a minimum by finding the second derivative. Taking the derivative of f'(x) gives us:
f''(x) = 2 + 2/x^3
Substituting x = 0.7937
f''(0.7937) = 2 + 2/(0.7937)^3 ≈ 4.1248
Since f''(0.7937) is positive, it confirms that x = 0.7937 is the number that minimizes the sum of its square and reciprocal.
So, the positive number that minimizes the expression x^2 + 1/x is approximately 0.7937.