What are the critical points of h(x)= 5x+(4/x) and are they minimum or maxmum values.

We have the function h(x) = 5x + (4/x). The critical
points of this function are at points where h'(x) = 0

h'(x)
= 5 - 4/x^2

5 - 4/x^2 =
0

=> x^2 =
4/5

=> x= 2/sqrt 5 and -2/sqrt
5

h''(x) = 8/x^3

At x = 2/sqrt
5, h''(x) is positive, therefore we have a local minimum
here.

h(2/sqrt 5) = 5*(2/ sqrt 5) + 4*sqrt 5 / 2 = 4*sqrt
5

At x = -2/sqrt 5, h''(x) is negative, therefore we have a
local minimum here.

h(2/sqrt 5) = 5*(2/ sqrt 5) + 4*sqrt 5
/ 2 = -4*sqrt 5

Here, we see that the local minimum has a
larger value than the local maximum, this is not an error. As the function does not have
an absolute maximum or an absolute minimum, the local extreme values can have the values
we have obtained.

The required point of local
minimum is ( 2/ sqrt 5 , 4*sqrt 5) and the point of local maximum is (-2/sqrt 5 ,
-4*sqrt 5)