how do i minimize this function.
21L + (2600/L)
Set the derivative equal to zero and solve for L.
21 - 2600/L^2 = 0
2600 = 21 L^2
There will be positive and negative solutions for L. You can use the second derivative test or direct calculation to see which gives a minimum.
To minimize the function 21L + (2600/L), you can follow these steps:
1. Take the derivative of the function with respect to L.
d/dL (21L + (2600/L)) = 21 - 2600/L^2
2. Set the derivative equal to zero and solve for L.
21 - 2600/L^2 = 0
Multiply both sides of the equation by L^2 to eliminate the denominator:
21L^2 - 2600 = 0
3. Solve this quadratic equation for L.
21L^2 = 2600
L^2 = 2600/21
L^2 ≈ 123.81
Take the square root of both sides to solve for L.
L ≈ ±11.13
So, there are two possible solutions for L: approximately 11.13 and -11.13.
4. To determine which solution gives a minimum, you can use the second derivative test or calculate the values of the function at L = 11.13 and L = -11.13.
Evaluating the function for each value:
F(L = 11.13) = 21(11.13) + (2600/11.13) ≈ 226.19
F(L = -11.13) = 21(-11.13) + (2600/-11.13) ≈ -226.19
Comparing these values, we can see that F(L = 11.13) gives a positive value (226.19), while F(L = -11.13) gives a negative value (-226.19).
Therefore, the minimum value of the function occurs when L ≈ 11.13, giving a minimum value of approximately 226.19.