Calculus homework help Optimization? Optimization Find a positive number x such that the sum of 4x and 1/xis?

Question

Calculus homework help Optimization?
Optimization
Find a positive number x such that the sum of 4x and 1/xis as small as possible.
what is x?

Answer 1

S = 4x + (1/x)

S = 4x + x^(-1)

dS/dx = 4 - x^(-2)

Set that to zero:

4 - x^(-2) = 0

x^(-2) = 4

x^2 = 1/4

x = +/- sqrt(1/4)

x = +/- 1/2

Since x is positive, x = 1/2 = 0.5

Answer 2

This question requires you to find the smallest positive number - so that's the smallest number in (0,∞) which is an open interval. To find this, we use differentiation. If you plot the graph of 4x + 1/x and find its minimum, that's where the value is lowest. y = 4x + 1/x thus dy/dx = 4 - 1/x² Setting this equal to zero and rearranging we obtain x² - 1/4 = 0 thus (x + 1/2)(x - 1/2) = 0 thus x = 1/2 or x = -1/2 As we require x to be positive, the answer is x = 1/2.

Answer 3

Ok---

Let f(x) = 4x + 1/x.

df/dx = 4-1/x^2 now set that = 0

4x^2 -1 = 0

solve for x

x = +/-0.5 and you wanted the positive number so it's 0.5.

If you take the 2nd derivative you'll find it to be positive so that indicates you have a minimum



You can also graph to check.

Answer 4

y = 4x + 1/x

dy/dx = 4 - 1/x^2

dy/dx = 0

0 = 4 - 1/x^2

1/x^2 = 4

x^2 = 1/4

x = +/- 1/2



y = 4 * (1/2) + 1/(1/2) = 2 + 2 = 4

y = 4 * (-1/2) + 1/(-1/2) = -2 - 2 = -4